Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[-1, 0], Q=[-12, -2]. 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=[-12, -2]. 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(-12, -1)
After
Length original PQ ORIGINAL Q minus P coordinate0 = -11
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(-2, 0)
After
Length original PQ ORIGINAL Q minus P coordinate1 = -2
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(-11, -11)
After
Length original PQ ORIGINAL square coordinate0 = 121
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(-2, -2)
After
Length original PQ ORIGINAL square coordinate1 = 4
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(121, 4)
After
Length original PQ ORIGINAL squared norm sum = 125
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], [-12, -2]]
After
derived = [[-11, -2], 125]
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
(-11, -2, 125)
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.