Question 7962: Two Scalar-Addition Routes and Component Defect

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=[-1, 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=[-1, 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

Step-by-step derivation

fixed standard basis;two ordered integer coordinates

componentwise operations;not length or inner product

a+b := sum(1, 2)

a+b = 3

(a+b)u[0] := mul(3, -1)

(a+b)u[0] = -3

(a+b)u[1] := mul(3, 2)

(a+b)u[1] = 6

au[0] := mul(1, -1)

au[0] = -1

au[1] := mul(1, 2)

au[1] = 2

bu[0] := mul(2, -1)

bu[0] = -2

bu[1] := mul(2, 2)

bu[1] = 4

au+bu[0] := sum(-1, -2)

au+bu[0] = -3

au+bu[1] := sum(2, 4)

au+bu[1] = 6

scalar additivity residual[0] := sub(-3, -3)

scalar additivity residual[0] = 0

scalar additivity residual[1] := sub(6, 6)

scalar additivity residual[1] = 0

a=1;b=2;u=[-1, 2]

left=[-3, 6];right=[-3, 6];residual=[0, 0]

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

([-1, 2], [-3, 6], [-3, 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.