# Question 7985: Ordered Weighted Vector Combination [Return to catalogue page 40](https://ocme.evemisslab.com/en/questions/page/40/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r01-vector-foundations/#q-20261012-advance-20261016-r01-vector-0010) 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. R0014-07985 · q-20261012-advance-20261016-r01-vector-0010 ## Original question and givens Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 7], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that order. - G001 — Use ordered integer two-component vectors in one fixed standard basis; zero scalars and zero vectors are allowed. Component arithmetic is not an inner product or length. u=[-3, 1], v=[1, 7], a=2, b=-1. Give (a*u, b*v, a*u+b*v) in that 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 fixed standard basis;two ordered integer coordinates After componentwise operations;not length or inner product Conditions: none Work coordinate by coordinate in the same fixed basis.Swapping coordinates,taking a norm or computing an inner product is not the requested vector answer. Step identifiers and dependenciesRule: ordered_vector_scope · Uses: G001 S002 · calculationBefore au[0] := mul(2, -3) After au[0] = -6 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore au[1] := mul(2, 1) After au[1] = 2 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore bv[0] := mul(-1, 1) After bv[0] = -1 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore bv[1] := mul(-1, 7) After bv[1] = -7 Conditions: none Multiply the specified scalar and component at this position;do not substitute an inner product or entrywise matrix product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore au+bv[0] := sum(-6, -1) After au+bv[0] = -7 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore au+bv[1] := sum(2, -7) After au+bv[1] = -5 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore au=[-6, 2];bv=[-1, -7] After value=[-7, -5] Conditions: none Each weight multiplies its own vector before aligned coordinates are added.Negative weights and the zero vector use the same definition. Step identifiers and dependenciesRule: weighted_vector_components · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute au+bv[1] with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute au+bv[0] with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute bv[1] with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute bv[0] with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute au[1] with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute au[0] with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([-6, 2], [-1, -7], [-7, -5]) 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: 16d96a9b8884e323de10789f45bf3977cdc5ece21764740885d18941d702afc9 - candidate record SHA-256: 659c43dec13e09058dbe3a42e2f1e197f9da95906ef3401f42d9fcba00763860 - semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77 - question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r01-vector-0010.json)