Vector Triple Product

Definition 6.5. For three vectors $\vec a,\vec b,\vec c$, the vector $\vec a\times(\vec b\times\vec c)$ is called a vector triple product. Given any three vectors, the genuine vector triple products are $(\vec a\times\vec b)\times\vec c,\ (\vec b\times\vec c)\times\vec a,\ (\vec c\times\vec a)\times\vec b$ and $\vec a\times(\vec b\times\vec c),\ \vec b\times(\vec c\times\vec a),\ \vec c\times(\vec a\times\vec b)$.

Theorem 6.7. The vector triple product is linear in each argument: $(\vec a_1+\vec a_2)\times(\vec b\times\vec c)=\vec a_1\times(\vec b\times\vec c)+\vec a_2\times(\vec b\times\vec c)$ and $(\lambda\vec a)\times(\vec b\times\vec c)=\lambda\big(\vec a\times(\vec b\times\vec c)\big)$, and similarly linear in the middle and last argument.

Theorem 6.8 (Vector Triple Product Expansion). For any three vectors $\vec a,\vec b,\vec c$:

This is proved by a clever choice of axes ($x$-axis along $\vec a$, $y$-axis in the plane of $\vec a,\vec b$, $z$-axis perpendicular to that plane) that makes all the cross products expand out to exactly this combination. It is THE workhorse formula of this section — it turns any vector triple product into two scalar (dot-product) coefficients times $\vec b$ and $\vec c$, with no cross product left to compute.

Companion facts (from Theorem 6.8 by relabelling):

1. $\vec a\times(\vec b\times\vec c)=\alpha\vec b+\beta\vec c$ where $\alpha=\vec a\cdot\vec c$ and $\beta=-\vec a\cdot\vec b$ — so it always lies in the plane parallel to $\vec b$ and $\vec c$ (never involves $\vec a$). …