A few points on implementation:
- The sub orthogonal is not very stable. Floating point numbers should be compared in terms of a small tolerance. A proper choice for two vectors could be pseudocoded as magnitude(v1-v2) < $EPS * sqrt(magnitude(v1)**2 + magnitude(v2)**2); $EPS is epsilon, the builtin precision of floats.
- Sub angle loses precision for angles near integer multiples of pi. An implementation in terms of atan2 would be better, extracting sin of the angle from the cross product.
- You may prefer to simply return the zero vector from normalize, instead of carping out.
- You'll find some help in the Math::Trig module. For high performance, take a look at Math::GSL and Math::Pari. If you decide to follow Masem's suggestion to generalize, PDL is another high performance library specializing in arrays of values.
I implemented some of the sexier parts of this stuff in the
snippet Cartesian 3-Vectors.
Update: Added #4. U^{2}: The pseudocode in #1 is for equality ( == operator). Other comparisons are similarly made.
After Compline,
Zaxo