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 another vector, not a number. The Cross product isn’t associative, so where you put the 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 $\mathbf{C}$ since it’s a linear combination of them. 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.

You can verify it component-wise. Worth memorizing because nested cross products show up in electromagnetics: the BAC-CAB form appears in 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 $\nabla$ playing the role of a vector. The first term on the right is "$\nabla$" (like $\mathbf{B}$) dotted with $\mathbf{F}$ (like $\mathbf{C}$) and multiplied back by another $\nabla$; the second term is the vector Laplacian. This is the identity you reach for when deriving the electromagnetic wave equation.