Question 29693: A Declared Real-Scalar Instance on Integer Coordinates

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. scalar=0, vector=[5, 0, -14]. Give (scalar,vector,scaled vector).

• 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. scalar=0, vector=[5, 0, -14]. Give (scalar,vector,scaled vector).

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

Step-by-step derivation

source roles = [qf-str22-vectorspace-scalar-multiplication, vectorspace-scalar-multiplication]

derived = [[scalar, vector, result]]

Vector ORIGINAL scalar product coordinate0 := mul(0, 5)

Vector ORIGINAL scalar product coordinate0 = 0

Vector ORIGINAL scalar product coordinate1 := mul(0, 0)

Vector ORIGINAL scalar product coordinate1 = 0

Vector ORIGINAL scalar product coordinate2 := mul(0, -14)

Vector ORIGINAL scalar product coordinate2 = 0

source roles = [0, [5, 0, -14]]

derived = [[0, 0, 0]]

Reverse checks

• S005 · 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.
• S004 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate2 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Vector ORIGINAL scalar product coordinate0 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, [5, 0, -14], [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.