Triple products JEE adds

The scalar triple product [a b c] = a·(b × c) is the determinant with rows (a₁, a₂, a₃), (b₁, b₂, b₃), (c₁, c₂, c₃). It is a number.

|[a b c]| is the volume of the parallelepiped with edges a, b, c from one corner. For edges 2î, î + 3ĵ, ĵ + 4k̂ the determinant is 2(12 − 0) = 24, so the volume is 24.

Three vectors are coplanar exactly when [a b c] = 0. For example î + ĵ, ĵ + k̂ and î + 2ĵ + k̂ are coplanar, since the third is the sum of the first two.

Cyclic order keeps the value, [a b c] = [b c a] = [c a b], and swapping two vectors flips its sign, as for a determinant. Also a·(b × c) = (a × b)·c.

Vector triple product: a × (b × c) = (a·c)b − (a·b)c. It lies in the plane of b and c, and in general a × (b × c) ≠ (a × b) × c.

Also useful: |a × b|² + (a·b)² = |a|²|b|² (Lagrange's identity), since sin²θ + cos²θ = 1.