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# 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 =

vx
vy

wx wy

vz wz
w = ψ ψ.

u=tw

tw, v

II.8

Given two linearly independent vectors u and v in three dimensions, their cross product is the vector uv 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

uv=

i

j k
u2 u3
v2 v3

uv = (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

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