Question 6529: Cyclic versus Swapped Signed Coordinate Crosses

Original question and givens

In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.

• G001 — In one equal-unit orthogonal Euclidean frame, labeled A,B,C have indices 1,2,3 and points=[[0, 0], [1, 0], [0, -9]]. The points are distinct and noncollinear. Give signed integers (crossABC, crossBCA, crossACB), distinguishing cyclic and swapped labels.

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-triangle

Step-by-step derivation

[[0, 0], [1, 0], [0, -9]]

original vertex labels1,2,3 in an equal-unit orthogonal Euclidean frame

[[0, 0], [1, 0], [0, -9]]

three distinct points=Yes

ABC first vector component0 := sub(1, 0)

ABC first vector component0 = 1

ABC first vector component1 := sub(0, 0)

ABC first vector component1 = 0

ABC second vector component0 := sub(0, 0)

ABC second vector component0 = 0

ABC second vector component1 := sub(-9, 0)

ABC second vector component1 = -9

ABC cross positive term := mul(1, -9)

ABC cross positive term = -9

ABC cross negative term := mul(0, 0)

ABC cross negative term = 0

ABC signed determinant := sub(-9, 0)

ABC signed determinant = -9

distinct=Yes;signed determinant=-9

proper triangle=Yes

BCA first vector component0 := sub(0, 1)

BCA first vector component0 = -1

BCA first vector component1 := sub(-9, 0)

BCA first vector component1 = -9

BCA second vector component0 := sub(0, 1)

BCA second vector component0 = -1

BCA second vector component1 := sub(0, 0)

BCA second vector component1 = 0

BCA cross positive term := mul(-1, 0)

BCA cross positive term = 0

BCA cross negative term := mul(-9, -1)

BCA cross negative term = 9

BCA signed determinant := sub(0, 9)

BCA signed determinant = -9

ACB first vector component0 := sub(0, 0)

ACB first vector component0 = 0

ACB first vector component1 := sub(-9, 0)

ACB first vector component1 = -9

ACB second vector component0 := sub(1, 0)

ACB second vector component0 = 1

ACB second vector component1 := sub(0, 0)

ACB second vector component1 = 0

ACB cross positive term := mul(0, 0)

ACB cross positive term = 0

ACB cross negative term := mul(-9, 1)

ACB cross negative term = -9

ACB signed determinant := sub(0, -9)

ACB signed determinant = 9

crossABC=-9;crossBCA=-9;crossACB=9

cyclic order preserves sign;one transposition reverses it

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute ACB signed determinant with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute ACB cross negative term with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute ACB cross positive term with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute ACB second vector component1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute ACB second vector component0 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute ACB first vector component1 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute ACB first vector component0 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute BCA signed determinant with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute BCA cross negative term with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute BCA cross positive term with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute BCA second vector component1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute BCA second vector component0 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute BCA first vector component1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute BCA first vector component0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute ABC signed determinant with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute ABC cross negative term with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute ABC cross positive term with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute ABC second vector component1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute ABC second vector component0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute ABC first vector component1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute ABC first vector component0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(-9, -9, 9)

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.