Question 7998: Ordered Weighted Vector Combination

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=[10, 1], v=[1, 20], 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=[10, 1], v=[1, 20], 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

Step-by-step derivation

fixed standard basis;two ordered integer coordinates

componentwise operations;not length or inner product

au[0] := mul(2, 10)

au[0] = 20

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

au[1] = 2

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

bv[0] = -1

bv[1] := mul(-1, 20)

bv[1] = -20

au+bv[0] := sum(20, -1)

au+bv[0] = 19

au+bv[1] := sum(2, -20)

au+bv[1] = -18

au=[20, 2];bv=[-1, -20]

value=[19, -18]

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

([20, 2], [-1, -20], [19, -18])

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.