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:
|w = ψ ψ.|
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 right-hand rule, with length equal to |u||v| sin Θ, where Θ is the angle between u and v. In rectangular coordinates, the cross product can be computed from the simple determi-nant formula
u ⋅ v = (u2v3 – u3v2, u3v1– u1v3, u1v2,– u2v1).
At first glance, cross product seems to be an artifact of three dimen-sions. 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