Find the projection of vec a×(vec b×vec c) onto vec b for…

Find the projection of $\vec a\times(\vec b\times\vec c)$ onto $\vec b$ for $\vec a=\{0,1,2\}$, $\vec b=\{2,1,-1\}$, $\vec c=\{2,1,1\}$.

Step 1: "BAC–CAB"

\[ \vec a\cdot\vec c=0+1+2=3,\qquad \vec a\cdot\vec b=0+1-2=-1, \]

\[ \vec a\times(\vec b\times\vec c)=3\vec b+\vec c=\{8,4,-2\}. \]

Step 2: Projection

\[ \frac{(3\vec b+\vec c)\cdot\vec b}{|\vec b|}=\frac{3\cdot6+4}{\sqrt6}=\frac{22}{\sqrt6}. \]

\[\mathrm{Pr}_{\vec b}\big[\vec a\times(\vec b\times\vec c)\big]=\frac{11\sqrt6}{3}\approx8.98\]

Direct check: $\vec b\times\vec c=\{2,-4,0\}$ and $\vec a\times\{2,-4,0\}=\{8,4,-2\}$ $\checkmark$. Then $\{8,4,-2\}\cdot\vec b=16+4+2=22$.