How do I use the given vector equation to resolve vector $p$ into a parallel and perpendicular component? I am working on the following problem: ![enter image description here]( Here's what I've done so far: ![enter image description here]( ![enter image description here]( I know that dotting the first component with q should equal one to show that it is parallel and dotting the second component with q should equal to 0 to show that it is equal to zero to show that it is perpendicular. I haven't been getting those two results. Please help!

The parallel component is obtained by the scalar product of $q$ with the dot product of $p$ and $q$, after normalization of $q$.

Hence $$\left(p\cdot\frac{q}{\|q\|}\right)\frac{q}{\|q\|}=\frac{p\cdot q}{\|q\|^2}q$$ i.e.

$$\frac{(3,-2,-1)\cdot(2,-2,3)}{2^2+(-2)^2+3^2}(2,-2,3)=\frac{7}{17}(2,-2,3).$$

The perpendicular component is the difference

$$(3,-2,-1)-\frac{7}{17}(2,-2,3).$$

If you multiply that by $q$, you get $7-7$.