How to generate vectors with given a certain vector and an angle?

Question

Let us start with 3-dimensional case.

Let $V_1=(X_1,Y_1,Z_1)$ and an angle $\theta$ be given, how to generate all the other possible vector $V_2=(X_2,Y_2,Z_2)$ such that $<V_1,V_2>=cos\theta$, where $<,>$ is the inner product. We only consider normalized vectors.

Is it possible to do it in higher dimensional space?

Thanks for helps!

Answer 1

Let x2 =s y2 = t then solve your equation and write z2 in terms of s and t. Varying s and t should generate all the vectors v2=(s , t , f (s,t)).