Vector triple product

The vector triple product of three 3D vectors is

$\mathbf{A}\times (\mathbf{B}\times \mathbf{C}),$

a vector. Unlike the Scalar triple product, the result is not a number but another vector — and the Cross product is not associative, so the placement of parentheses matters.

BAC-CAB identity

$\mathbf{A}\times (\mathbf{B}\times \mathbf{C})=\mathbf{B}(\mathbf{A}\cdot \mathbf{C})-\mathbf{C}(\mathbf{A}\cdot \mathbf{B}).$

Mnemonic: “BAC minus CAB.” The result lies in the plane of $\mathbf{B}$ and $\mathbf{C}$ (because it’s a linear combination of them), which makes sense geometrically — $\mathbf{B}\times \mathbf{C}$ is perpendicular to that plane, and $\mathbf{A}\times (\text{perpendicular})$ brings you back into the plane.

The identity can be verified component-wise. It’s worth memorizing because nested cross products show up in electromagnetics (the BAC-CAB form appears in calculations of force on a moving charge, in expansions of $\mathbf{E}\times \mathbf{B}$, and in the curl-of-curl identity).

Non-associativity

$\mathbf{A}\times (\mathbf{B}\times \mathbf{C})\ne (\mathbf{A}\times \mathbf{B})\times \mathbf{C}$

in general. Each side is a different vector in a different plane:

• $\mathbf{A}\times (\mathbf{B}\times \mathbf{C})$ lies in the plane of $\mathbf{B}$ and $\mathbf{C}$.
• $(\mathbf{A}\times \mathbf{B})\times \mathbf{C}$ lies in the plane of $\mathbf{A}$ and $\mathbf{B}$.

Always write the parentheses explicitly when working with iterated cross products.

Connection to vector calculus identities

The identity $\nabla \times (\nabla \times \mathbf{F})=\nabla (\nabla \cdot \mathbf{F})-{\nabla }^2\mathbf{F}$ has the same BAC-CAB structure with the operators $\nabla$ playing the role of vectors. The first term on the right is "$\nabla$" (acting like $\mathbf{B}$) dotted with $\mathbf{F}$ (like $\mathbf{C}$) and multiplied back by another $\nabla$; the second term is the vector Laplacian. This identity is foundational in electromagnetic wave-equation derivations.