# Question 29668: One Exact Instance of Distributivity over Scalar Addition [Return to catalogue page 149](https://ocme.evemisslab.com/en/questions/page/149/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r04-real-vector-space-foundations/#q-20261013-advance-20261022-r04-real-vector-space-0093) 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. R0058-29668 · q-20261013-advance-20261022-r04-real-vector-space-0093 ## Original question and givens Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=0, vector=[5, 0, -15]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?). - G001 — Use a fixed real coordinate space with nonempty vectors of the same dimension; integer coordinates and scalars are special real instances. Check only the declared coordinate instance, not all abstract vector-space axioms for an arbitrary structure. a=0, b=0, vector=[5, 0, -15]. Give (a,b,vector,(a+b)*vector,a*vector+b*vector,equal?). ## 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-real-vector-space](https://ocme.evemisslab.com/en/mko/mko-real-vector-space/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-vectorspace-scalar-distributivity, vectorspace-scalar-distributivity] After derived = [[a, b, vector, left, right, equal]] Conditions: none Follow THIS bank exact keys,domain/frame,operation and answer role.STR22/App20 tuples differ from legacy scalars;retain or omit/1 by the ORIGINAL contract,not a generic visual convention. Step identifiers and dependenciesRule: R58_EXACT_source_interface_and_answer_ROLE · Uses: G001 S002 · calculationBefore Vector ORIGINAL scalar a plus b := sum(0, 0) After Vector ORIGINAL scalar a plus b = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Vector ORIGINAL left distributed coordinate0 := mul(0, 5) After Vector ORIGINAL left distributed coordinate0 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Vector ORIGINAL a term coordinate0 := mul(0, 5) After Vector ORIGINAL a term coordinate0 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Vector ORIGINAL b term coordinate0 := mul(0, 5) After Vector ORIGINAL b term coordinate0 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Vector ORIGINAL right distributed coordinate0 := sum(0, 0) After Vector ORIGINAL right distributed coordinate0 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Vector ORIGINAL left distributed coordinate1 := mul(0, 0) After Vector ORIGINAL left distributed coordinate1 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Vector ORIGINAL a term coordinate1 := mul(0, 0) After Vector ORIGINAL a term coordinate1 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Vector ORIGINAL b term coordinate1 := mul(0, 0) After Vector ORIGINAL b term coordinate1 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Vector ORIGINAL right distributed coordinate1 := sum(0, 0) After Vector ORIGINAL right distributed coordinate1 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Vector ORIGINAL left distributed coordinate2 := mul(0, -15) After Vector ORIGINAL left distributed coordinate2 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Vector ORIGINAL a term coordinate2 := mul(0, -15) After Vector ORIGINAL a term coordinate2 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Vector ORIGINAL b term coordinate2 := mul(0, -15) After Vector ORIGINAL b term coordinate2 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Vector ORIGINAL right distributed coordinate2 := sum(0, 0) After Vector ORIGINAL right distributed coordinate2 = 0 Conditions: none Substitute exact ORIGINAL operands.Additional teaching algebra does not inherit raw-input caps;only declared SOURCE rational operations apply their own profile. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · impliesBefore source roles = [0, 0, [5, 0, -15]] After derived = [[0, 0, 0], [0, 0, 0], Yes] Conditions: none Separately calculate(a+b)*x_i and a*x_i+b*x_i for EVERY coordinate;real distributivity equates them.The derived a+b may exceed raw100 without clipping.This shows the declared coordinate instance and its reason. Step identifiers and dependenciesRule: Vector_distributivity_TWO_complete_coordinate_routes · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. - S014 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Vector ORIGINAL right distributed coordinate0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Vector ORIGINAL b term coordinate0 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Vector ORIGINAL a term coordinate0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Vector ORIGINAL left distributed coordinate0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Vector ORIGINAL scalar a plus b with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check EACH original datum,index,direction,representative and premise,not only the answer.Exact candidate checking,independent review,formal proof and publication clearance remain separate. ## Result (0, 0, [5, 0, -15], [0, 0, 0], [0, 0, 0], Yes) 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: 868808197ea45e834b104eae17e2692dc8f42a60aa3b4d88823accdec608c2ac - candidate record SHA-256: 7f146ad95f7dab68ccb5fa34246f6f591e7dd1a04843b611b88768ea6f55407d - semantic review receipt SHA-256: 66bbb31c5fe644576dfd47ac71ca9e367fc23b948eac8e3eb648d96dadea05df - question-first witness SHA-256: b1e4fbdbca8d0ac3581e2f26466374d870f3c8498fab7cf6873feca5bbec44e7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r04-real-vector-space-0093.json)