Sum of parallel and perpendicular vectors

Question

If $v_1 + v_2 = \langle-5,5\rangle$ where
$v_1$ is parallel to $\langle-3,5\rangle$ and
$v_2$ is perpendicular to $\langle-3,5\rangle$.
Then what are the two vectors. I’m not quite sure how I should approach this question

Answer 1

First find a vector perpendicular to $(-3,5)$. You want a vector that has a dot product of zero with this vector. Inspection should quickly show that $(5,3)$ works fine.

Now set $\lambda(-3,5) + \mu(5,3)= (-5,5)$, where $\lambda$ and $\mu$ are scalars.

By splitting the components, you can now get two simple linear simultaneous equations ($-3\lambda + 5\mu = -5$ and $5\lambda + 3\mu = 5$) to solve for the scalars.