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:

Here's what I've done so far:


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!


I know that dotting the first component with q should equal one to
  show that it is parallel

Not in general:
$$
p_\parallel \cdot q 
= \lVert p_\parallel \rVert \lVert q \rVert \cos \angle(p_\parallel, q)
= \lVert p_\parallel \rVert
$$

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.

Check your calculation:
$$
(p \times q)_1 = -6/\sqrt{17} - (-1 \cdot -2/\sqrt{17}) = -8 / \sqrt{17}
$$
etc.
