Prove that (vec a+vec b)·[(vec a+vec c)×vec b]=(vec a×vec…

Prove that $(\vec a+\vec b)\cdot\big[(\vec a+\vec c)\times\vec b\big]=(\vec a\times\vec c)\cdot\vec b$.

Step 1: Expansion

\[ (\vec a+\vec b)\cdot(\vec a\times\vec b+\vec c\times\vec b)=[\vec a,\vec a,\vec b]+[\vec a,\vec c,\vec b]+[\vec b,\vec a,\vec b]+[\vec b,\vec c,\vec b], \] using $\vec u\cdot(\vec v\times\vec w)=[\vec u,\vec v,\vec w]$ (Problem 109).

Step 2: Repeated vectors vanish

A triple product with two equal vectors is $0$, since $\vec v\times\vec w$ is perpendicular to $\vec v$ and to $\vec w$. Three of the four terms disappear: $=[\vec a,\vec c,\vec b]=(\vec a\times\vec c)\cdot\vec b.\qquad\blacksquare$.

\[(\vec a+\vec b)\cdot\big[(\vec a+\vec c)\times\vec b\big]=(\vec a\times\vec c)\cdot\vec b\]