Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[8, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
G001 — Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[1, 0], Q=[8, -5]. Give (Qx−Px, Qy−Py, L2(P,Q)²), keeping the first two signs.
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.
Retain the ORIGINAL bank full operation prefix,domain and ordered answer roles.Use orthogonal_equal_unit only where a frame is declared,and degrees where an angle unit is declared.Integer fractions retain/1.Do not invent parameters or alter original bilingual sources.
Length original PQ ORIGINAL Q minus P coordinate0 := sub(8, 1)
After
Length original PQ ORIGINAL Q minus P coordinate0 = 7
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Length original PQ ORIGINAL Q minus P coordinate1 := sub(-5, 0)
After
Length original PQ ORIGINAL Q minus P coordinate1 = -5
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Length original PQ ORIGINAL square coordinate0 := mul(7, 7)
After
Length original PQ ORIGINAL square coordinate0 = 49
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Length original PQ ORIGINAL square coordinate1 := mul(-5, -5)
After
Length original PQ ORIGINAL square coordinate1 = 25
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
Length original PQ ORIGINAL squared norm sum := sum(49, 25)
After
Length original PQ ORIGINAL squared norm sum = 74
Conditions: none
Substitute exact ORIGINAL roles,without floating approximation.SAFE bounds raw inputs only;derived integers/rationals are unbounded and do not inherit another bank component cap.
source roles = [Length original PQ, [1, 0], [8, -5]]
After
derived = [[7, -5], 74]
Conditions: none
In the SAME equal-unit orthogonal L2 frame,the directed displacement is Q-P with signs retained.The sum of coordinate squares is the nonnegative SQUARED length.Derived coordinates may exceed SAFE without being revalidated as raw inputs.
S007 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
S006 · premise_and_result_audit — Recompute Length original PQ ORIGINAL squared norm sum with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute Length original PQ ORIGINAL square coordinate1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute Length original PQ ORIGINAL square coordinate0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute Length original PQ ORIGINAL Q minus P coordinate1 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Recompute Length original PQ ORIGINAL Q minus P coordinate0 with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check this step against the ORIGINAL data,indices and premises,without claiming the conclusion uniquely recovers the inputs.Finite exact checking,independent review,formal proof and publication clearance remain separate.
Result
(7, -5, 74)
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.