Question 7927: Both Directed Vector Differences and Their Sum

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, -1], v=[1, -1]. Give (u, v, u−v, v−u, sum of the two differences).

• 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, -1], v=[1, -1]. Give (u, v, u−v, v−u, sum of the two differences).

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

u-v[0] := sub(1, 1)

u-v[0] = 0

u-v[1] := sub(-1, -1)

u-v[1] = 0

v-u[0] := sub(1, 1)

v-u[0] = 0

v-u[1] := sub(-1, -1)

v-u[1] = 0

opposite differences sum[0] := sum(0, 0)

opposite differences sum[0] = 0

opposite differences sum[1] := sum(0, 0)

opposite differences sum[1] = 0

u=[1, -1];v=[1, -1];u-v=[0, 0];v-u=[0, 0]

sum=[0, 0]

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 opposite differences sum[1] with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute opposite differences sum[0] with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute v-u[1] with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute v-u[0] with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute u-v[1] with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute u-v[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

([1, -1], [1, -1], [0, 0], [0, 0], [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.