Part of V3D-03 — Vectors: Advanced (Triple Product, Coplanarity)

Scalar Triple Product — Definition and Computation

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The scalar triple product (STP) of vectors a, b, c is defined as [a b c] = a.(b x c).

If a = (a1,a2,a3), b = (b1,b2,b3), c = (c1,c2,c3), then:

[a b c] = |a1 a2 a3; b1 b2 b3; c1 c2 c3| (3x3 determinant with a, b, c as rows).

Computation: expand the determinant along the first row: a1(b2c3-b3c2) - a2(b1c3-b3c1) + a3(b1c2-b2c1).

Equivalent forms: [a b c] = a.(b x c) = b.(c x a) = c.(a x b) = (a x b).c.

The STP can also be computed as the determinant with a, b, c as columns (same result, since det(M) = det(MTM^T)).

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