Question

How to find vector S that is orthogonal to vector V and at an known angle from vector Z.

Answer 1

S·V = 0, S·Z = cos(θ). This is a system of two linear equations in the three unknown components of S, which can be solved to yield a generic line of pseudo-solutions (pseudo because most won't be unitary and won't form an angle θ with Z). Further restricting these to unit vectors will yield finitely many true solutions (the intersection of the line with the unit sphere).

Answer 2

Heres my 2 pence:

Let Y = Z cross V

Y is perpendicular to both Z and V

To get S, we need to add something to Y, in the Z direction, call it W = alpha * Z, such that:

Z dot S = cos theta, ie

Z dot (W+Y) = cos theta

Z dot (alpha * Z + Y) = cos theta

alpha * Z dot Z + Z dot Y = cos theta

alpha * 1 + 0 = cos theta

alpha = cos theta

S = Y + cos theta * Z