Question 31414: Squared Euclidean Distance

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(1, -8) and Q=(-12, 6). Find d(P,Q)² for the Euclidean L2 distance.

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

source roles = [qf-distance-squared, squared_distance, frame, euclidean_orthonormal_equal_units]

derived = [[integer_squared_norm]]

Distance PQ ORIGINAL Q minus P axis0 := sub(-12, 1)

Distance PQ ORIGINAL Q minus P axis0 = -13

Distance PQ ORIGINAL Q minus P axis1 := sub(6, -8)

Distance PQ ORIGINAL Q minus P axis1 = 14

Distance PQ ORIGINAL signed delta squared axis0 := mul(-13, -13)

Distance PQ ORIGINAL signed delta squared axis0 = 169

Distance PQ ORIGINAL signed delta squared axis1 := mul(14, 14)

Distance PQ ORIGINAL signed delta squared axis1 = 196

Distance PQ ORIGINAL sum of two squares := sum(169, 196)

Distance PQ ORIGINAL sum of two squares = 365

source roles = [Distance PQ, [1, -8], [-12, 6]]

derived = [[-13, 14], [169, 196], 365]

Reverse checks

• S007 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
• S006 · premise_and_result_audit — Recompute Distance PQ ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Distance PQ ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Distance PQ ORIGINAL Q minus P axis0 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.

Result

365

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.