the question says: Find a unit vector that is orthogonal to a= <-3,5>
I know that two vectors are orthogonal iff a dot b =0, but I'm not sure how to find the answer if it is a unit vector.
In two dimensions, you can find a vector orthogonal to a given one by switching the coordinates and changing the sign of one of them. So a vector orthogonal to (–3,5) is (5,3).
That's not a unit vector, but you can convert it into one by dividing by its length, which is $\displaystyle \sqrt{5^2+3^2} = \sqrt{34}$.
So a unit vector orthogonal to (–3,5) is $\displaystyle \bigl(\tfrac5{\sqrt{34}},\tfrac3{\sqrt{34}}\bigr)$.