Question 31366: Exact Comparison of Two Euclidean Distances

Original question and givens

Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). In the same frame, U=(-10, -18) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > in that order.

• G001 — Use one equal-unit orthonormal Euclidean Cartesian frame. P=(3, -10) and Q=(-12, 6). In the same frame, U=(-10, -18) and V=(5, -2). Compare d(P,Q) and d(U,V), giving <, =, or > 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-distance-formula

Step-by-step derivation

source roles = [qf-distance-comparison, distance_comparison, frame, euclidean_orthonormal_equal_units]

derived = [[PQ_versus_UV_operator]]

Distance PQ ORIGINAL Q minus P axis0 := sub(-12, 3)

Distance PQ ORIGINAL Q minus P axis0 = -15

Distance PQ ORIGINAL Q minus P axis1 := sub(6, -10)

Distance PQ ORIGINAL Q minus P axis1 = 16

Distance PQ ORIGINAL signed delta squared axis0 := mul(-15, -15)

Distance PQ ORIGINAL signed delta squared axis0 = 225

Distance PQ ORIGINAL signed delta squared axis1 := mul(16, 16)

Distance PQ ORIGINAL signed delta squared axis1 = 256

Distance PQ ORIGINAL sum of two squares := sum(225, 256)

Distance PQ ORIGINAL sum of two squares = 481

source roles = [Distance PQ, [3, -10], [-12, 6]]

derived = [[-15, 16], [225, 256], 481]

Distance UV ORIGINAL Q minus P axis0 := sub(5, -10)

Distance UV ORIGINAL Q minus P axis0 = 15

Distance UV ORIGINAL Q minus P axis1 := sub(-2, -18)

Distance UV ORIGINAL Q minus P axis1 = 16

Distance UV ORIGINAL signed delta squared axis0 := mul(15, 15)

Distance UV ORIGINAL signed delta squared axis0 = 225

Distance UV ORIGINAL signed delta squared axis1 := mul(16, 16)

Distance UV ORIGINAL signed delta squared axis1 = 256

Distance UV ORIGINAL sum of two squares := sum(225, 256)

Distance UV ORIGINAL sum of two squares = 481

source roles = [Distance UV, [-10, -18], [5, -2]]

derived = [[15, 16], [225, 256], 481]

source roles = [481, 481]

derived = [=]

Reverse checks

• S014 · 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.
• S013 · 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.
• S012 · premise_and_result_audit — Recompute Distance UV ORIGINAL sum of two squares with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Distance UV ORIGINAL signed delta squared axis1 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Distance UV ORIGINAL signed delta squared axis0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Distance UV ORIGINAL Q minus P axis1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Distance UV ORIGINAL Q minus P axis0 with exact rational arithmetic, not rounded decimals.
• 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

=

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.