{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0001","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca209a2e1ca5ff779194abcb98e7f5ced5bbc1140257982c2ef6df4fc1e123fb","family_id":"qf-vector-coordinate-addition","seed":202610045001,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0002","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ece9c9afe2ae5bf0e9761dc34492ab18afd2e796e26e2089cf06e65a7ad1891","family_id":"qf-vector-coordinate-addition","seed":202610045002,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-11,-1],\"v\":[11,-2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-11, -1) and v=(11, -2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -3)","explanation":"Apply addition componentwise to obtain (0, -3). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0003","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db87fc268950c5084bebcc313779536e54ea088f954b752cd25cd8f4fce56edf","family_id":"qf-vector-coordinate-addition","seed":202610045003,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-10,0],\"v\":[10,-1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-10, 0) and v=(10, -1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -1)","explanation":"Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0004","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dc2e88182ecff28acf212c172f8e28f183168b8587c9dbd482cce55b5dbff59","family_id":"qf-vector-coordinate-addition","seed":202610045004,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-9,1],\"v\":[9,0]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-9, 1) and v=(9, 0). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 1)","explanation":"Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0005","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3569d6d9beac8849b795b6f6a96a2e3247fdf2768d00859a634cd19f752d974d","family_id":"qf-vector-coordinate-addition","seed":202610045005,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-8,2],\"v\":[8,1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-8, 2) and v=(8, 1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 3)","explanation":"Apply addition componentwise to obtain (0, 3). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0006","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff64e26fa45c51c1083e2243a4c8ec44981f1d4095ece61c1271ac0d02ec20cf","family_id":"qf-vector-coordinate-addition","seed":202610045006,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-7,-2],\"v\":[7,2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-7, -2) and v=(7, 2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0007","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5293959f66be38f850228ea8c9f31b6e86e5e2b7bd5e6092f4094578ccbfdc20","family_id":"qf-vector-coordinate-addition","seed":202610045007,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-6,-1],\"v\":[6,3]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-6, -1) and v=(6, 3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 2)","explanation":"Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0008","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38cf6ddfb159b33ab6095dad9a19ca1d42b99788e18a877ac34a5a3ca8f533bf","family_id":"qf-vector-coordinate-addition","seed":202610045008,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-5,0],\"v\":[5,-3]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-5, 0) and v=(5, -3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -3)","explanation":"Apply addition componentwise to obtain (0, -3). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0009","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c60594d3cb537994373a0d99825199d16af90f30711eeb9f89bd0dbbbf53e04","family_id":"qf-vector-coordinate-addition","seed":202610045009,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-4,1],\"v\":[4,-2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-4, 1) and v=(4, -2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -1)","explanation":"Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0010","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f50778bfbad29a54a6e25370b63c407c348134f20ece29df66e61acfce7fdb6f","family_id":"qf-vector-coordinate-addition","seed":202610045010,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-3,2],\"v\":[3,-1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-3, 2) and v=(3, -1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 1)","explanation":"Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0011","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b64b37e2e5a59b0af66848eaafabbaf628862879bec2b8d963af197fda5dce2","family_id":"qf-vector-coordinate-addition","seed":202610045011,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-2,-2],\"v\":[2,0]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-2, -2) and v=(2, 0). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -2)","explanation":"Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0012","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39a126c01b5e9299584dd5eb28cd93b2014a9be4af7b8e6db681b1a684f76715","family_id":"qf-vector-coordinate-addition","seed":202610045012,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-1,-1],\"v\":[1,1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-1, -1) and v=(1, 1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0013","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb1789ccd3cdeb1b4bf6f7867a8d33f573cee79bdd2d66a3bd3eba5e8d578b30","family_id":"qf-vector-coordinate-addition","seed":202610045013,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[0,0],\"v\":[0,2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and v=(0, 2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 2)","explanation":"Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0014","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fabc7b6b883ad884478371bb4aaf3126bfa0d9adad4006fcd1028e23cf25d3d7","family_id":"qf-vector-coordinate-addition","seed":202610045014,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-1,3],\"v\":[1,1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(1, 1) and v=(-1, 3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 4)","explanation":"Apply addition componentwise to obtain (0, 4). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0015","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1689473bf275671a11d22fb22f1972e47282c957f04a1f3ffba801a4e0001956","family_id":"qf-vector-coordinate-addition","seed":202610045015,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-2,-3],\"v\":[2,2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(2, 2) and v=(-2, -3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -1)","explanation":"Apply addition componentwise to obtain (0, -1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0016","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d924be5266eca6c100b03bae355fa795a0851567a1eb126ec2f09ec1759baaa","family_id":"qf-vector-coordinate-addition","seed":202610045016,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-3,-2],\"v\":[3,-2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(3, -2) and v=(-3, -2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -4)","explanation":"Apply addition componentwise to obtain (0, -4). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0017","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12f1f948184b77bf21c955b5eef471315f581715e22cb8a91c9726c736dfacc4","family_id":"qf-vector-coordinate-addition","seed":202610045017,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-4,-1],\"v\":[4,-1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(4, -1) and v=(-4, -1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -2)","explanation":"Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0018","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b1c5c6d5ee11267b198d433790570a954a8202e4bd0156e89bbd62c42e424de","family_id":"qf-vector-coordinate-addition","seed":202610045018,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-5,0],\"v\":[5,0]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(5, 0) and v=(-5, 0). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0019","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0b7fdf5ed337096ce2c5680da762d844f2285fbc3479173882a972327d30810","family_id":"qf-vector-coordinate-addition","seed":202610045019,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-6,1],\"v\":[6,1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(6, 1) and v=(-6, 1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 2)","explanation":"Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0020","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"031ec12133694e5aeded4952864144059045d60ee5ac7108640a36c199a1d379","family_id":"qf-vector-coordinate-addition","seed":202610045020,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-7,2],\"v\":[7,2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(7, 2) and v=(-7, 2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 4)","explanation":"Apply addition componentwise to obtain (0, 4). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0021","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b53e17acbd58762be98090a2d9e863852d3a49f68eecf17ddcb6365247b9a87","family_id":"qf-vector-coordinate-addition","seed":202610045021,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-8,3],\"v\":[8,-2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(8, -2) and v=(-8, 3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 1)","explanation":"Apply addition componentwise to obtain (0, 1). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0022","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd9151917750da2ff6d563baec370423247e58baae7613bc2e6efcfa4cd9b7e1","family_id":"qf-vector-coordinate-addition","seed":202610045022,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-9,-3],\"v\":[9,-1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(9, -1) and v=(-9, -3). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -4)","explanation":"Apply addition componentwise to obtain (0, -4). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0023","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78571401f6bea49a44a260c4a6f22b95887c8a1799c253eea0fab8b25e9a0ff6","family_id":"qf-vector-coordinate-addition","seed":202610045023,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-10,-2],\"v\":[10,0]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(10, 0) and v=(-10, -2). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, -2)","explanation":"Apply addition componentwise to obtain (0, -2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0024","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4fb22154b1251c3251ffd00cab65482b40e742e62b3fd37f4bc9698971f216b","family_id":"qf-vector-coordinate-addition","seed":202610045024,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-11,-1],\"v\":[11,1]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(11, 1) and v=(-11, -1). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Apply addition componentwise to obtain (0, 0). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0025","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47cb7be3eb53a3358073f94807bf760235508e6f70a658e72f2ba9b9e77da5a5","family_id":"qf-vector-coordinate-addition","seed":202610045025,"semantic_template":"qf-vector-coordinate-addition/{\"domain\":\"reals\",\"operation\":\"vector_add\",\"u\":[-12,0],\"v\":[12,2]}"},"fields":{"title":"Componentwise Vector Addition in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(12, 2) and v=(-12, 0). Compute the exact components of u+v.","answer":{"kind":"expression","value":"(0, 2)","explanation":"Apply addition componentwise to obtain (0, 2). These are vector components, not a geometric point or a length."},"skills":["Componentwise Vector Addition in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0026","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd09613ffd0216ef04ffc034dc5b666b8af14b48c82844de62e319ffeb288cb8","family_id":"qf-vector-scalar-multiplication","seed":202610045026,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[0,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and scalar k=0. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0027","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84778917e339d56077971a46f29511f6d4e0cbb4d3c866b62578b08be571c321","family_id":"qf-vector-scalar-multiplication","seed":202610045027,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[-11,-1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-11, -1) and scalar k=-2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(22, 2)","explanation":"Multiply both components by k to obtain ku=(22, 2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0028","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0111d6564179e368cf1f50382e2878138bc653e00b6636b2d723d9c00af1d46c","family_id":"qf-vector-scalar-multiplication","seed":202610045028,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[-10,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-10, 0) and scalar k=-1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(10, 0)","explanation":"Multiply both components by k to obtain ku=(10, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0029","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fa572411b9b3e6c732e2728d7b3b34733917a23a249862f6a6251b4a522b5cd","family_id":"qf-vector-scalar-multiplication","seed":202610045029,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[-9,1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-9, 1) and scalar k=0. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0030","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b00d92e8e3ad03632bcc3ea707274759cc42ce2b900df76497ff67ce348f687","family_id":"qf-vector-scalar-multiplication","seed":202610045030,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[-8,2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-8, 2) and scalar k=1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-8, 2)","explanation":"Multiply both components by k to obtain ku=(-8, 2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0031","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f98ad007ccec53396e7e808e32389c8810300be2bfcf95c73da17ba3efa3edc","family_id":"qf-vector-scalar-multiplication","seed":202610045031,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[-7,-2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-7, -2) and scalar k=2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-14, -4)","explanation":"Multiply both components by k to obtain ku=(-14, -4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0032","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"763d752ab971f72926e8bc5cdf79e2765300a54fbe6048ebe13ad480587a8d05","family_id":"qf-vector-scalar-multiplication","seed":202610045032,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[-6,-1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-6, -1) and scalar k=3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-18, -3)","explanation":"Multiply both components by k to obtain ku=(-18, -3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0033","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83bc8dc5132b83b7f4952a45efaad8f93e449a3cb900923d0ecc6a2e37c56f43","family_id":"qf-vector-scalar-multiplication","seed":202610045033,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[-5,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-5, 0) and scalar k=-3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(15, 0)","explanation":"Multiply both components by k to obtain ku=(15, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0034","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a150f8c5ed40e28a46cc2116df99cc95097ecb87e35e3e1a043d1401a7e6a7d","family_id":"qf-vector-scalar-multiplication","seed":202610045034,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[-4,1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-4, 1) and scalar k=-2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(8, -2)","explanation":"Multiply both components by k to obtain ku=(8, -2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0035","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4835366c1a07d2c575e35cd0f3bc1c6b56747d9fda517ca35ab23228e86816a6","family_id":"qf-vector-scalar-multiplication","seed":202610045035,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[-3,2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-3, 2) and scalar k=-1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(3, -2)","explanation":"Multiply both components by k to obtain ku=(3, -2). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0036","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fb5e86e08cfff01990d9e56f9a03ec9f37dfddea45ef6883e2687c1844773b55","family_id":"qf-vector-scalar-multiplication","seed":202610045036,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[-2,-2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-2, -2) and scalar k=0. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0037","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b048756b3a8e195ecce4f26e6eed9f0352058e6563357c33fb2ea936ecaed3d5","family_id":"qf-vector-scalar-multiplication","seed":202610045037,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[-1,-1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-1, -1) and scalar k=1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-1, -1)","explanation":"Multiply both components by k to obtain ku=(-1, -1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0038","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4064486d5e150a8650e367c23718622f7eded358d1d206ca7f8493f1c560abc4","family_id":"qf-vector-scalar-multiplication","seed":202610045038,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[0,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and scalar k=2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0039","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a0468b29bcce7ee5f86362d8b375f8ce7ceb7c6c7473806144990425c3caf0","family_id":"qf-vector-scalar-multiplication","seed":202610045039,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[1,1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(1, 1) and scalar k=3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(3, 3)","explanation":"Multiply both components by k to obtain ku=(3, 3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0040","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd890fb4c705aedb217210163ddb72837f8ccda3825ec996d6c4dff29eb2d942","family_id":"qf-vector-scalar-multiplication","seed":202610045040,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[2,2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(2, 2) and scalar k=-3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-6, -6)","explanation":"Multiply both components by k to obtain ku=(-6, -6). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0041","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d14bff843bdc4f259a063615cb8da616216ca9bffff785d9772fbd4029ca036","family_id":"qf-vector-scalar-multiplication","seed":202610045041,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[3,-2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(3, -2) and scalar k=-2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-6, 4)","explanation":"Multiply both components by k to obtain ku=(-6, 4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0042","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efe89e3b2c6419b45d99313dad4c94187b1c68b6bee024049c49aae54c834f05","family_id":"qf-vector-scalar-multiplication","seed":202610045042,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[4,-1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(4, -1) and scalar k=-1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-4, 1)","explanation":"Multiply both components by k to obtain ku=(-4, 1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0043","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"396d4f8ebb9a5ba2a84b48f4c45fd556f30bf79a9c805380eab35ed7804cbc12","family_id":"qf-vector-scalar-multiplication","seed":202610045043,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[5,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(5, 0) and scalar k=0. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0044","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cfaa03cde25e678da9c180b7251c65976780d9e7103c910224baaf0097f8eea0","family_id":"qf-vector-scalar-multiplication","seed":202610045044,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":1,\"operation\":\"scalar_multiply\",\"u\":[6,1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(6, 1) and scalar k=1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(6, 1)","explanation":"Multiply both components by k to obtain ku=(6, 1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0045","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eaa885080e6cabd9382d27ec3ce9a7c9f89e1fe01358c0829c6c91943b684c29","family_id":"qf-vector-scalar-multiplication","seed":202610045045,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":2,\"operation\":\"scalar_multiply\",\"u\":[7,2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(7, 2) and scalar k=2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(14, 4)","explanation":"Multiply both components by k to obtain ku=(14, 4). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0046","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e07a9251126653b1b171ad27a7be53e14f928f99ef47ff14b540e3bf86df312d","family_id":"qf-vector-scalar-multiplication","seed":202610045046,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":3,\"operation\":\"scalar_multiply\",\"u\":[8,-2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(8, -2) and scalar k=3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(24, -6)","explanation":"Multiply both components by k to obtain ku=(24, -6). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0047","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a69384b39312e7470d1cf05f69970a9aef9a0b1c9900221bfe73f20c92432be","family_id":"qf-vector-scalar-multiplication","seed":202610045047,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-3,\"operation\":\"scalar_multiply\",\"u\":[9,-1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(9, -1) and scalar k=-3. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-27, 3)","explanation":"Multiply both components by k to obtain ku=(-27, 3). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0048","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c9780b932edbf86283a680c687cdd3be2e121653ad98d5c2300e188413914ef","family_id":"qf-vector-scalar-multiplication","seed":202610045048,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-2,\"operation\":\"scalar_multiply\",\"u\":[10,0]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(10, 0) and scalar k=-2. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-20, 0)","explanation":"Multiply both components by k to obtain ku=(-20, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0049","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28111cc42374f26bcbac9c9c9d44ee82741def9a798efe327f43f10158f520fb","family_id":"qf-vector-scalar-multiplication","seed":202610045049,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":-1,\"operation\":\"scalar_multiply\",\"u\":[11,1]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(11, 1) and scalar k=-1. Compute the exact components of ku.","answer":{"kind":"expression","value":"(-11, -1)","explanation":"Multiply both components by k to obtain ku=(-11, -1). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0050","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a255b02c8f00bda22870b923ab6784b243573cb3181e3f1f85e25411a8afa19d","family_id":"qf-vector-scalar-multiplication","seed":202610045050,"semantic_template":"qf-vector-scalar-multiplication/{\"domain\":\"reals\",\"k\":0,\"operation\":\"scalar_multiply\",\"u\":[12,2]}"},"fields":{"title":"Scalar Multiplication in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(12, 2) and scalar k=0. Compute the exact components of ku.","answer":{"kind":"expression","value":"(0, 0)","explanation":"Multiply both components by k to obtain ku=(0, 0). A zero scalar gives the zero vector; scalar multiplication is not an inner product and implies no length calculation."},"skills":["Scalar Multiplication in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0051","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c06ae03e06e1b051abb9a445a338a2087bfc5118fa8fcc4a39586cdab573c0d0","family_id":"qf-vector-coordinate-difference","seed":202610045051,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-12,-2],\"v\":[-3,12]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-12, -2) and v=(-3, 12). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -14)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -14). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0052","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"958bf2ae977391d1dc12feef9da67022a0ca92473a8e77a5fd60c4a096afbf52","family_id":"qf-vector-coordinate-difference","seed":202610045052,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-11,-1],\"v\":[-2,11]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-11, -1) and v=(-2, 11). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -12)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -12). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0053","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7415a4b3d353737433d50c9b5934d135bc26154285fab8c75bb8bd9757418d99","family_id":"qf-vector-coordinate-difference","seed":202610045053,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-10,0],\"v\":[-1,10]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-10, 0) and v=(-1, 10). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -10)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -10). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0054","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd24e9b52b14b981dbca86bf01eb2c462e1a3f5147ee58c8d6c1e8501c56dbef","family_id":"qf-vector-coordinate-difference","seed":202610045054,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-9,1],\"v\":[0,9]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-9, 1) and v=(0, 9). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -8)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -8). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0055","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b3083e647e3eb4bd88db46b5a0b9c353b36400fbd955fceaacf1a92a6d492b0","family_id":"qf-vector-coordinate-difference","seed":202610045055,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-8,2],\"v\":[1,8]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-8, 2) and v=(1, 8). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -6)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -6). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0056","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c51afaf0fa34d354456f2eb7a625a35eda873b551d742fe14b0e4758ed33777e","family_id":"qf-vector-coordinate-difference","seed":202610045056,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-7,-2],\"v\":[2,7]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-7, -2) and v=(2, 7). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -9)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -9). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0057","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"900bd1d5ab34b94124ce251cbb970ac76d0f441ff80b377122c65be6c8bbee3a","family_id":"qf-vector-coordinate-difference","seed":202610045057,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-6,-1],\"v\":[3,6]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-6, -1) and v=(3, 6). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-9, -7)","explanation":"Apply ordered subtraction componentwise to obtain (-9, -7). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0058","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba6700ac8553b8a946a5aa8e48081e6aba3cc9b08e2ed4bb9b23ca597be70a7a","family_id":"qf-vector-coordinate-difference","seed":202610045058,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-5,0],\"v\":[-3,5]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-5, 0) and v=(-3, 5). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, -5)","explanation":"Apply ordered subtraction componentwise to obtain (-2, -5). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0059","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"12a3cf1ac1cb620c615473f34dc41e235df0ce47f2567f07983c99f8db29e894","family_id":"qf-vector-coordinate-difference","seed":202610045059,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-4,1],\"v\":[-2,4]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-4, 1) and v=(-2, 4). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, -3)","explanation":"Apply ordered subtraction componentwise to obtain (-2, -3). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0060","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0023ec44134170654ad28d805e8c851de0e36558d4de3c9705fe510046128df2","family_id":"qf-vector-coordinate-difference","seed":202610045060,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-3,2],\"v\":[-1,3]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-3, 2) and v=(-1, 3). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, -1)","explanation":"Apply ordered subtraction componentwise to obtain (-2, -1). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0061","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20c70fc472f014fb4a1905500d3f088de6045420d48d1d5eb4bf3908f13d0779","family_id":"qf-vector-coordinate-difference","seed":202610045061,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-2,-2],\"v\":[0,2]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-2, -2) and v=(0, 2). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, -4)","explanation":"Apply ordered subtraction componentwise to obtain (-2, -4). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0062","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a075ba6f91b2d3c495713032ea597ff177fa44f98fac1646b4ebba94f939d8d4","family_id":"qf-vector-coordinate-difference","seed":202610045062,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[-1,-1],\"v\":[1,1]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-1, -1) and v=(1, 1). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, -2)","explanation":"Apply ordered subtraction componentwise to obtain (-2, -2). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0063","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a73c287dcb629a0fe51d39d352ab90f26a6a6b1f26edcc5ab2f18d4a3134f538","family_id":"qf-vector-coordinate-difference","seed":202610045063,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[0,0],\"v\":[2,0]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and v=(2, 0). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, 0)","explanation":"Apply ordered subtraction componentwise to obtain (-2, 0). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0064","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a2331c46f77286c60fc454567b06a5bfb72a93f26e46f237e41971f98d780b3","family_id":"qf-vector-coordinate-difference","seed":202610045064,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[1,1],\"v\":[3,-1]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(1, 1) and v=(3, -1). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(-2, 2)","explanation":"Apply ordered subtraction componentwise to obtain (-2, 2). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0065","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"378751d86f819a95569e5bcf09a210272067de3b1af3322aa3e5aaa5f2f66da8","family_id":"qf-vector-coordinate-difference","seed":202610045065,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[2,2],\"v\":[-3,-2]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(2, 2) and v=(-3, -2). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 4)","explanation":"Apply ordered subtraction componentwise to obtain (5, 4). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0066","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b48a4ec233bea75a59e2b8ec279d9b9ffc1238136b7ab57f0a5faedd7478fe1","family_id":"qf-vector-coordinate-difference","seed":202610045066,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[3,-2],\"v\":[-2,-3]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(3, -2) and v=(-2, -3). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 1)","explanation":"Apply ordered subtraction componentwise to obtain (5, 1). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0067","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e6e46d5dead9da7869a05fbb7b944f1bf9865b8cfb654932e944923f5b25b3e","family_id":"qf-vector-coordinate-difference","seed":202610045067,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[4,-1],\"v\":[-1,-4]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(4, -1) and v=(-1, -4). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 3)","explanation":"Apply ordered subtraction componentwise to obtain (5, 3). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0068","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32e4a0829c7222fd9775d17c0aaf1640b45f414ffa7dc8ff9c84884a1296bd16","family_id":"qf-vector-coordinate-difference","seed":202610045068,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[5,0],\"v\":[0,-5]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(5, 0) and v=(0, -5). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 5)","explanation":"Apply ordered subtraction componentwise to obtain (5, 5). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0069","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b4d5818adc8ddfee4be68fab7932c36a746ec226875599630346e4e5ee99edc","family_id":"qf-vector-coordinate-difference","seed":202610045069,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[6,1],\"v\":[1,-6]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(6, 1) and v=(1, -6). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 7)","explanation":"Apply ordered subtraction componentwise to obtain (5, 7). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0070","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8edd68a1f53583255631b5c164dd428f2d1d68358b131dd8764dac585241348c","family_id":"qf-vector-coordinate-difference","seed":202610045070,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[7,2],\"v\":[2,-7]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(7, 2) and v=(2, -7). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 9)","explanation":"Apply ordered subtraction componentwise to obtain (5, 9). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0071","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db7090c2f320427504d71008d8d0d5bc437f7d9859830b3ac997efc71ae1c5e0","family_id":"qf-vector-coordinate-difference","seed":202610045071,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[8,-2],\"v\":[3,-8]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(8, -2) and v=(3, -8). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(5, 6)","explanation":"Apply ordered subtraction componentwise to obtain (5, 6). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0072","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a02c2076ed969d04611b03c558ae55bc5e910e8aa6f0b7b24ec461b1261895f","family_id":"qf-vector-coordinate-difference","seed":202610045072,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[9,-1],\"v\":[-3,-9]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(9, -1) and v=(-3, -9). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(12, 8)","explanation":"Apply ordered subtraction componentwise to obtain (12, 8). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0073","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d021d4ab3feb45267120517f71b330a86296c4854a0da5d09e45aa4830cd20d","family_id":"qf-vector-coordinate-difference","seed":202610045073,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[10,0],\"v\":[-2,-10]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(10, 0) and v=(-2, -10). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(12, 10)","explanation":"Apply ordered subtraction componentwise to obtain (12, 10). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0074","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6b335a64ca55b11de7b5d0883cb3749a2089eb3409450cbec4a6b04bcfedc5f","family_id":"qf-vector-coordinate-difference","seed":202610045074,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[11,1],\"v\":[-1,-11]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(11, 1) and v=(-1, -11). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(12, 12)","explanation":"Apply ordered subtraction componentwise to obtain (12, 12). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0075","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed89672e5b098668c8534e35f1d9bb6ff63b021db0500b3544d3553746234452","family_id":"qf-vector-coordinate-difference","seed":202610045075,"semantic_template":"qf-vector-coordinate-difference/{\"domain\":\"reals\",\"operation\":\"vector_subtract\",\"u\":[12,2],\"v\":[0,-12]}"},"fields":{"title":"Ordered Vector Subtraction in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(12, 2) and v=(0, -12). Compute the exact components of u−v, preserving the subtraction order.","answer":{"kind":"expression","value":"(12, 14)","explanation":"Apply ordered subtraction componentwise to obtain (12, 14). These are vector components, not a geometric point or a length."},"skills":["Ordered Vector Subtraction in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0076","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52aa7e1aa876d414e0d1dd2426a8a0b6301578df4ccd0b7c8696bb5db932b807","family_id":"qf-vector-coordinate-equality","seed":202610045076,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[0,0],\"v\":[0,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 0) and v=(0, 0). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0077","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d736242b1f6ca60163e37e0fcca75035fe8394cd39dfaa32bba9efe0b1d4ae5","family_id":"qf-vector-coordinate-equality","seed":202610045077,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-10,-2],\"v\":[-11,-1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-11, -1) and v=(-10, -2). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0078","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790c8f15ed994bec9058ede457e5c945a01c498537c32f7433fbd8115a265554","family_id":"qf-vector-coordinate-equality","seed":202610045078,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-10,0],\"v\":[-10,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-10, 0) and v=(-10, 0). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0079","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e80a6613dc0651cf56b50dd72e9a4c7fb4e6abd8fcf1b67c49cadc189c3f5c4b","family_id":"qf-vector-coordinate-equality","seed":202610045079,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-9,1],\"v\":[1,-9]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-9, 1) and v=(1, -9). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0080","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72fa6d7683ff14ee31f05d0cd5d75d48a1f0c2d0753670f9124172af8338a9e3","family_id":"qf-vector-coordinate-equality","seed":202610045080,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-8,2],\"v\":[-8,2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-8, 2) and v=(-8, 2). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0081","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5939a3d5e99896b6fef7fde7ae7b4ecf106f7ba908acfc7379279b3724280fd2","family_id":"qf-vector-coordinate-equality","seed":202610045081,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-6,-3],\"v\":[-7,-2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-7, -2) and v=(-6, -3). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0082","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c75c6e7137329e1c324de0d88863e50eeb9edf1d97e94f1c41fccb792aa0bc03","family_id":"qf-vector-coordinate-equality","seed":202610045082,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-6,-1],\"v\":[-6,-1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-6, -1) and v=(-6, -1). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0083","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d8bc23980f42b617ff4d8a7fe2ecae3ea2a935c8497b6aee95b18a651406912","family_id":"qf-vector-coordinate-equality","seed":202610045083,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-4,-1],\"v\":[-5,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-5, 0) and v=(-4, -1). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0084","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"759931bb86b618536eb90fb874bdc61b566f407a9359c6e45fb198acb4e38afd","family_id":"qf-vector-coordinate-equality","seed":202610045084,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-4,1],\"v\":[-4,1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-4, 1) and v=(-4, 1). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0085","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67a6e8c795d650486b0b9c018a336c32a68e7dca8a89c6092122199be873873d","family_id":"qf-vector-coordinate-equality","seed":202610045085,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-2,1],\"v\":[-3,2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-3, 2) and v=(-2, 1). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0086","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40c0c9bba7818105c761857d9626033209dc1027262ff72f53bf24545e5d5f18","family_id":"qf-vector-coordinate-equality","seed":202610045086,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-2,-2],\"v\":[-2,-2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-2, -2) and v=(-2, -2). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0087","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ae434475d03278db6e48fdfc88d83de9681d9ee2c6a7ca82395ac9e9b2594db","family_id":"qf-vector-coordinate-equality","seed":202610045087,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[-1,-1],\"v\":[0,-2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(-1, -1) and v=(0, -2). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0088","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7994fc8e21e8936f2867fcec85df603fb263a271b730344aa95231696682fc9b","family_id":"qf-vector-coordinate-equality","seed":202610045088,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[0,4],\"v\":[0,4]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(0, 4) and v=(0, 4). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0089","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b49456a12ba88fd61b0420203c8766424182dcf311d7719510933bffb1989e2","family_id":"qf-vector-coordinate-equality","seed":202610045089,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[1,1],\"v\":[2,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(1, 1) and v=(2, 0). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0090","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea35cd130d6cce73977c815d3ac0f47ebada091f3ea97305e721f59e18f11934","family_id":"qf-vector-coordinate-equality","seed":202610045090,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[2,2],\"v\":[2,2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(2, 2) and v=(2, 2). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0091","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4800e3bc32872abf88a4a202e8a3ed62b5f05ff0e8e9638971f1164f5307af8d","family_id":"qf-vector-coordinate-equality","seed":202610045091,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[3,-2],\"v\":[4,-3]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(3, -2) and v=(4, -3). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0092","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abf6f329a3f0b10fead4874608be6c322e389d6f021d3fc675a00d7fe992dc8e","family_id":"qf-vector-coordinate-equality","seed":202610045092,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[4,-1],\"v\":[4,-1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(4, -1) and v=(4, -1). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0093","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01471176b81b389672af6220e7fa8ffeb37f5a71151286e6ee6632bd077b314b","family_id":"qf-vector-coordinate-equality","seed":202610045093,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[5,0],\"v\":[6,-1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(5, 0) and v=(6, -1). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0094","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d12d0e75e9dc2e8c0a43705a09df8d6d3640205c78ceb9297bdee58790cede91","family_id":"qf-vector-coordinate-equality","seed":202610045094,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[6,1],\"v\":[6,1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(6, 1) and v=(6, 1). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0095","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f64a90488a67ec9dbc55014e278a96780f7b79d76f307a42319cd8b86e5be0d2","family_id":"qf-vector-coordinate-equality","seed":202610045095,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[7,2],\"v\":[8,1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(7, 2) and v=(8, 1). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0096","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8592b1aba73c2592f700d698fee85795b22c4b142fe96cfd4be6f47a32bb5b3","family_id":"qf-vector-coordinate-equality","seed":202610045096,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[8,-2],\"v\":[8,-2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(8, -2) and v=(8, -2). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0097","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2c4b4e7a1f700289180f65c463c671aea749f153cb75104f4bf54f3ad62d8f5","family_id":"qf-vector-coordinate-equality","seed":202610045097,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[10,-2],\"v\":[9,-1]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(9, -1) and v=(10, -2). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0098","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df66de0fa98be900f7610f54892270df27c70f8c58334852e312ef7e1164b677","family_id":"qf-vector-coordinate-equality","seed":202610045098,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[10,0],\"v\":[10,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(10, 0) and v=(10, 0). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0099","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f4a8000f423800600ab10846d2049278647da62801c7e72697bd1f8f96cb411","family_id":"qf-vector-coordinate-equality","seed":202610045099,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[11,1],\"v\":[12,0]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(11, 1) and v=(12, 0). Are u and v equal?","answer":{"kind":"short_text","value":"No","explanation":"Equality requires both corresponding components to agree, giving No. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r05-vector-0100","batch_id":"batch-20261003-advance-20261004-r05-vector-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb4064df38826a8d4ac322fe0d15ce5d8ad3d201cf437883a6fb5e002b63540e","family_id":"qf-vector-coordinate-equality","seed":202610045100,"semantic_template":"qf-vector-coordinate-equality/{\"domain\":\"reals\",\"operation\":\"vector_equality\",\"u\":[12,2],\"v\":[12,2]}"},"fields":{"title":"Vector Equality in a Fixed Basis","stem":"In the fixed basis of the real plane R², u=(12, 2) and v=(12, 2). Are u and v equal?","answer":{"kind":"short_text","value":"Yes","explanation":"Equality requires both corresponding components to agree, giving Yes. Equality of component sums or arbitrary exchange of components is insufficient."},"skills":["Vector Equality in a Fixed Basis"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}