How to calculate a vector from an angle with another vector in 2D?

Question

I know I should know this, but I just can't figure it out/find the solution. I can do it by hand, but I can't put it in an algorithm... (working in c++, but pseudocode should be fine).

I have a vector, and I want to find another vector based on the angle with it.

v is known, angle alpha is known and the magnitude of w is known. How can I find w?

Thanks!

Answer 1

To rotate a vector v = (x, y) by an angle alpha clockwise about the origin, you can multiply by the matrix:

[  cos alpha    sin alpha ]
[ -sin alpha    cos alpha ]

Thus the rotated vector with the same magnitude will be

(x cos alpha + y sin alpha, -x sin alpha + y cos alpha).

To change the magnitude from |v| to |w|, multiply both co-ordinates by |w|/|v|.