Question

How do I find a third vector in the same plane as the first two vectors but at a angle from one of the vectors

Answer 1

since they are all coplanar vectors, the third vector can be assumed as xi+yj where i and j are unit vectors in the direction of X, Y axes respectively. now you can use the definition of DOT product and solve for x and y.

Answer 2

then rotate the vectors as needed to align with the known vectors

X = xcos(\(\theta\))-ysin(\(\theta\))
Y = xsin\(\theta\))+ycos(\(\theta\))