Question 4051: Ordered Pair of Squared Euclidean Distances

Original question and givens

In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]] in order. Give their two squared L2 distances.

• G001 — In one equal-unit orthogonal Euclidean plane, L2 is the nonnegative distance from the common origin to each vector. The vectors are [[0, 0], [3, 4]] in order. Give their two squared L2 distances.

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-distance-formula

Step-by-step derivation

integer vectors from a common origin;orthogonal equal-length units

L2(v)≥0 and L2(v)²=v₁²+v₂²

vector1 coordinate1 squared := square(0)

vector1 coordinate1 squared = 0

vector1 coordinate2 squared := square(0)

vector1 coordinate2 squared = 0

vector1 L2 squared := sum(0, 0)

vector1 L2 squared = 0

vector2 coordinate1 squared := square(3)

vector2 coordinate1 squared = 9

vector2 coordinate2 squared := square(4)

vector2 coordinate2 squared = 16

vector2 L2 squared := sum(9, 16)

vector2 L2 squared = 25

Reverse checks

• S007 · premise_and_result_audit — Recompute vector2 L2 squared with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute vector2 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute vector2 coordinate1 squared with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute vector1 L2 squared with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute vector1 coordinate2 squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute vector1 coordinate1 squared 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

(0, 25)

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.