Scalar triple product

The scalar triple product

a \cdot (b \times c)

is the signed volume of the parallelepiped spanned by three vectors.

Table of contents

Intersection parameters from scalar triple products ( $s = [a, b, n] / [a, m, n], t = [a, b, m] / [a, m, n]$ )

Consider lines

r = a + s m

and

r = b + t n

with nonzero direction vectors. If they intersect, then

a - b = t n - s m

, so the necessary coplanarity condition is

[a - b, m, n] = 0

. For nonparallel directions it is also sufficient, since the orthogonal complement of

m \times n

is the linear span of

m, n

. For parallel directions the condition alone is insufficient: distinct parallel lines also satisfy it.

Assume intersection and

D = [a, m, n] \neq = 0

. Take the scalar triple products

[a, b, n]

and

[a, b, m]

of

b = a + s m - t n

. The scalar triple product is an alternating multilinear map, giving respectively

sD

and

t D

, proving the title formulas. The nonzero denominator fixes a convenient determinant chart; intersections with

D = 0

can still exist and require a different auxiliary vector or another method.

Four-vector scalar triple product identity ( $[b, c, d] a - [a, c, d] b + [a, b, d] c - [a, b, c] d = 0$ )

For four vectors in three-dimensional real Euclidean space, write

[a, b, c] = a \cdot (b \times c)

. Expand the same cross product in two ways using the vector triple product identity:

(a \times b) \times (c \times d) = [a, b, d] c - [a, b, c] d = [a, c, d] b - [b, c, d] a .

Rearrangement gives the displayed identity. It expresses linear dependence of four vectors in three dimensions with explicit determinant coefficients and remains valid when some or all of those coefficients vanish.

Cross products with a common vector ( $(A \times B) \times (A \times C) = (A \cdot (B \times C)) A$ )

The cross product of two cross products sharing their first vector is parallel to that vector, with scalar factor equal to the scalar triple product. In suffix notation, use

ϵ_{ijk} ϵ_{k pq} = δ_{i p} δ_{j q} - δ_{i q} δ_{j p}

. The component of the double cross product becomes

A_{i} ϵ_{q ℓ m} C_{q} A_{ℓ} B_{m} - C_{i} ϵ_{p ℓ m} A_{p} A_{ℓ} B_{m}

. The second term vanishes by antisymmetry of the Levi-Civita symbol; the first equals

A_{i} (A \cdot (B \times C))

, proving the identity even when the vectors are dependent.

Squared scalar triple product from cyclic cross products ( $(a \times b) \cdot [(b \times c) \times (c \times a)] = [a \cdot (b \times c)]^{2}$ )

Apply the vector triple product identity with first vector

b \times c

. Its contraction with

c

vanishes, so

(b \times c) \times (c \times a) = [a \cdot (b \times c)] c

. Dotting with

a \times b

gives the square of the scalar triple product. This is also the Gram determinant of

a, b, c

, hence is nonnegative even when the vectors are linearly dependent.

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