# Question 35940: Ordered Vector Subtraction in a Fixed Basis [Return to catalogue page 180](https://ocme.evemisslab.com/en/questions/page/180/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261003-advance-20261004-r05-vector-foundations/#q-20261003-advance-20261004-r05-vector-0065) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0070-35940 · q-20261003-advance-20261004-r05-vector-0065 ## Original question and givens 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. - G001 — 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. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-vector](https://ocme.evemisslab.com/en/mko/mko-vector/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-vector-coordinate-difference, [domain, operation, u, v]] After derived = [vector, vector_subtract, reals] Conditions: none Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules. Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001 S002 · impliesBefore source roles = [[2, 2], [-3, -2], null] After derived = [two real coordinates in ONE ordered basis] Conditions: none The vectors use ONE ordered basis.Type ALL TWO RAW SAFE components and k.These are vector coordinates,not points,lengths or a dot product.Component sums alone do not determine equality. Step identifiers and dependenciesRule: Vector_GIVEN_common_ordered_basis_RAW_two_SAFE_components · Uses: G001, S001 S003 · calculationBefore Vector ORIGINAL coordinate i=0 := sub(2, -3) After Vector ORIGINAL coordinate i=0 = 5 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Vector ORIGINAL coordinate i=1 := sub(2, -2) After Vector ORIGINAL coordinate i=1 = 4 Conditions: none This step displays the exact original operands and operation.Mul/sub/div have EXACTLY TWO inputs.Keep rational values exact and derived BigInts unbounded;do not force an answer by approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [[2, 2], [-3, -2], null, vector_subtract] After derived = [[5, 4], (5, 4)] Conditions: none Operate componentwise in ORIGINAL order.u−v must not reverse;zero scaling yields the zero vector.The result is an ORDERED tuple,not a set or length.Derived components may exceed RAW SAFE. Step identifiers and dependenciesRule: Vector_ORDERED_componentwise_result · Uses: G001, S004 ## Reverse checks - S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S004 · premise_and_result_audit — Recompute Vector ORIGINAL coordinate i=1 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Vector ORIGINAL coordinate i=0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. - S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved. ## Result (5, 4) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 378751d86f819a95569e5bcf09a210272067de3b1af3322aa3e5aaa5f2f66da8 - candidate record SHA-256: f151748a5b450459b2f7d9575f1bb8b91a0a6f70fc6795e2091e0ae57c922aef - semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80 - question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261003-advance-20261004-r05-vector-0065.json)