# Question 30128: Directed Components and Squared Length [Return to catalogue page 151](https://ocme.evemisslab.com/en/questions/page/151/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r06-euclidean-length-foundations/#q-20261008-advance-20261012-r06-euclidean-length-0003) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0059-30128 · q-20261008-advance-20261012-r06-euclidean-length-0003 ## Original question and givens Use one equal-unit orthogonal Euclidean L2 plane and one length unit. P=[0, -1], Q=[-10, 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=[0, -1], Q=[-10, 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. - [mko-euclidean-length](https://ocme.evemisslab.com/en/mko/mko-euclidean-length/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density12-length-displacement-square, density12_length_displacement_square, integer_euclidean_plane] After derived = [[DIRECTED_Qx_minus_Px, DIRECTED_Qy_minus_Py, SQUARED_length]] Conditions: none 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. Step identifiers and dependenciesRule: R59_EXACT_family_domain_FULLprefix_operation_and_answer_roles · Uses: G001 S002 · calculationBefore Length original PQ ORIGINAL Q minus P coordinate0 := sub(-10, 0) After Length original PQ ORIGINAL Q minus P coordinate0 = -10 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Length original PQ ORIGINAL Q minus P coordinate1 := sub(2, -1) After Length original PQ ORIGINAL Q minus P coordinate1 = 3 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Length original PQ ORIGINAL square coordinate0 := mul(-10, -10) After Length original PQ ORIGINAL square coordinate0 = 100 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Length original PQ ORIGINAL square coordinate1 := mul(3, 3) After Length original PQ ORIGINAL square coordinate1 = 9 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Length original PQ ORIGINAL squared norm sum := sum(100, 9) After Length original PQ ORIGINAL squared norm sum = 109 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [Length original PQ, [0, -1], [-10, 2]] After derived = [[-10, 3], 109] 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. Step identifiers and dependenciesRule: Euclidean_DIRECTED_displacement_then_SQUARED_nonnegative_norm · Uses: G001, S006 ## Reverse checks - 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 (-10, 3, 109) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 658979096141eba892b514b6496dedf6d58e97345a0eda1b220db5b74b02a042 - candidate record SHA-256: 298f6ed2aecfdf1719dab9e5b6e13b8e5abe2706a7843e9ce71e5380e24fb421 - semantic review receipt SHA-256: 4c93dea3956f58509e20b3b8bbbf716dbc340b424a2a5177bc04aba9e8ea9ec5 - question-first witness SHA-256: a5e56e0d64d622d7202840cef7019839c5a99244c602694190779155207dafdf - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r06-euclidean-length-0003.json)