For vec a=\1,2,1\ , vec b=\1,-1,2\ , vec c=\2,1,-1\ find vec…

For $\vec a=\{1,2,1\}$, $\vec b=\{1,-1,2\}$, $\vec c=\{2,1,-1\}$ find $\vec a\times(\vec b\times\vec c)$ and the projection of $\vec b$ onto it.

Step 1: Direct computation

\[ \vec b\times\vec c=\begin{vmatrix}\vec i&\vec j&\vec k\\1&-1&2\\2&1&-1\end{vmatrix}=\{-1,5,3\}, \]

\[ \vec a\times(\vec b\times\vec c)=\begin{vmatrix}\vec i&\vec j&\vec k\\1&2&1\\-1&5&3\end{vmatrix}=\{1,-4,7\}. \]

\[\vec p=\vec a\times(\vec b\times\vec c)=\{1,-4,7\}\]

Step 2: Check with the "BAC–CAB" rule

\[ \vec a\times(\vec b\times\vec c)=\vec b\,(\vec a\cdot\vec c)-\vec c\,(\vec a\cdot\vec b)=3\vec b-1\cdot\vec c=\{3,-3,6\}-\{2,1,-1\}=\{1,-4,7\}.\ \checkmark \]

($\vec a\cdot\vec c=2+2-1=3$, $\vec a\cdot\vec b=1-2+2=1$.) The result lies in the plane of $\vec b$ and $\vec c$.

Step 3: Projection of $\vec b$

\[ \vec b\cdot\vec p=1+4+14=19,\qquad |\vec p|=\sqrt{1+16+49}=\sqrt{66}. \]

\[\mathrm{Pr}_{\vec p}\vec b=\frac{19}{\sqrt{66}}=\frac{19\sqrt{66}}{66}\approx2.34\]