Their cross product the vector perpendicular the plane and
II.7 NONUNIFORM RANDOM POINTS VIA WARPING
Typically, we want a directional (θ, φ) pair to be with respect tosomeunit vector y (as opposed to the z axis). To do this we can first convert the angles to a unit vector a:
R = 




vz  wz 
w = ψ ψ. 

u=t⋅w t⋅w, v 

II.8
Given two linearly independent vectors u and v in three dimensions, their cross product is the vector u⋅v perpendicular to the plane of u and v, oriented according to the righthand rule, with length equal to uv sin Θ, where Θ is the angle between u and v. In rectangular coordinates, the cross product can be computed from the simple determinant formula
u⋅v= 

j  k  

u2  u3  
v2  v3 
u ⋅ v = (u2v3 – u3v2, u3v1– u1v3, u1v2,– u2v1).
At first glance, cross product seems to be an artifact of three dimensions. In three dimensions the normal direction to the plane determined by two vectors is unique up to sign, but in four dimensions there are a whole plane of vectors normal to any given plane. Thus, it is unclear how to define the cross product of two vectors in four dimensions. What then