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