Compute the projection of vec a=\3,-12,4\ onto vec b=vec…

Compute the projection of $\vec a=\{3,-12,4\}$ onto $\vec b=\vec c\times\vec d$, where $\vec c=\{1,0,-2\}$ and $\vec d=\{1,3,-4\}$.

Explanation of Method

The projection of a onto the vector product of c and d.

Step 1: The vector $\vec b$

\[ \vec b=\begin{vmatrix}\vec i&\vec j&\vec k\\1&0&-2\\1&3&-4\end{vmatrix}=\vec i(0+6)-\vec j(-4+2)+\vec k(3-0)=\{6,2,3\},\qquad |\vec b|=\sqrt{49}=7. \]

Step 2: Projection

\[ \vec a\cdot\vec b=18-24+12=6,\qquad \mathrm{Pr}_{\vec b}\vec a=\frac{6}{7}. \]

\[\mathrm{Pr}_{\vec b}\vec a=\frac67\]

Note that $\vec a\cdot(\vec c\times\vec d)=[\vec a,\vec c,\vec d]=6$ is a scalar triple product. With $|\vec a|=13$, the vector $\vec a$ is almost parallel to the plane of $\vec c,\vec d$.

$O$ $\vec c$ $\vec d$ $\vec b$

$\vec b=\vec c\times\vec d=\{6,2,3\},\ |\vec b|=7$

$\vec b$ is normal to the plane of $\vec c$ and $\vec d$ (shaded parallelogram). The projection of $\vec a$ onto $\vec b$ is only $\tfrac67$, so $\vec a$ (length 13) is almost parallel to that plane.

Result

\[\boxed{\,\mathrm{Pr}_{\vec b}\vec a=\frac67\,}\]