Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-2, -1], Q=[6, 0]. 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=[-2, -1], Q=[6, 0]. 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(6, -2)
After
Length original PQ ORIGINAL Q minus P coordinate0 = 8
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(0, -1)
After
Length original PQ ORIGINAL Q minus P coordinate1 = 1
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(8, 8)
After
Length original PQ ORIGINAL square coordinate0 = 64
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(1, 1)
After
Length original PQ ORIGINAL square coordinate1 = 1
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(64, 1)
After
Length original PQ ORIGINAL squared norm sum = 65
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, [-2, -1], [6, 0]]
After
derived = [[8, 1], 65]
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
(8, 1, 65)
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.