equation of a 3rd vector that is coplanar to 2 other vectors that are at right angles

Hi folks!

I start with 2 vectors (in 3 dimensions) that are at right angles to each other (and are therefore coplanar). I want to find a third vector that lies in the same plane as the other two but at a specific angle to one of the vectors ( eg. 23 degrees to vector 2 ).

Another way to think of the problem is that I want to change the angle of one of the two vectors to something other than 90 degrees but to ensure that it remains coplanar with the original two vectors.

So to clarify - I have 3 knowns: 2 vectors and 1 angle; and I have 1 unknown: 1 vector in the same plane as the 2 other vectors at an angle of X to one of the other vectors. I'm looking for a solution to calculate the unknown.

My first thoughts were that the new vector is a line on the surface of a right cone formed with it's axis as one of the first vectors; and that one of the solutions to the intersection of the cone and the plane formed by the first two vectors will be the answer. I'm not sure how to do this mathematically - and perhaps there's a better solution.

Can someone with more math skill than I please help?

Thanks!