# Question 7966: Two Scalar-Addition Routes and Component Defect [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-0066) 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. R0014-07966 · q-20261012-advance-20261016-r01-vector-0066 ## 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, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes). - 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, 2], a=1, b=2. Give (u, (a+b)u, au+bu, difference of the last two routes). ## 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 a+b := sum(1, 2) After a+b = 3 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore (a+b)u[0] := mul(3, 3) After (a+b)u[0] = 9 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 (a+b)u[1] := mul(3, 2) After (a+b)u[1] = 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, S003 S005 · calculationBefore au[0] := mul(1, 3) After au[0] = 3 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[1] := mul(1, 2) 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, S005 S007 · calculationBefore bu[0] := mul(2, 3) After bu[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, S006 S008 · calculationBefore bu[1] := mul(2, 2) After bu[1] = 4 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, S007 S009 · calculationBefore au+bu[0] := sum(3, 6) After au+bu[0] = 9 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore au+bu[1] := sum(2, 4) After au+bu[1] = 6 Conditions: none Add the already aligned component or position,without mixing coordinates. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore scalar additivity residual[0] := sub(9, 9) After scalar additivity residual[0] = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore scalar additivity residual[1] := sub(6, 6) After scalar additivity residual[1] = 0 Conditions: none Subtract the second operand from the first in the requested order,retaining the residual sign. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore a=1;b=2;u=[3, 2] After left=[9, 6];right=[9, 6];residual=[0, 0] Conditions: none Adding scalars first and scaling separately are two routes.Check that each residual component is zero for these supplied values. Step identifiers and dependenciesRule: two_scalar_routes_audit · Uses: G001, S012 ## Reverse checks - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Recompute scalar additivity residual[1] with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute scalar additivity residual[0] with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute au+bu[1] with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute au+bu[0] with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute bu[1] with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute bu[0] with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute au[1] with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute au[0] with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute (a+b)u[1] with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute (a+b)u[0] with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute a+b 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 ([3, 2], [9, 6], [9, 6], [0, 0]) 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: 0efa95f67516b8fd78f26a82077f0e95ba246811eef6f7497685f957b6e0079c - candidate record SHA-256: 0315703cffc246b5128c78e9d518f848798453378b65241842e4c4989f2d76cd - 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-0066.json)