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(v1v2) < $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 3Vectors.
Update: Added #4. U^{2}: The pseudocode in #1 is for equality ( == operator). Other comparisons are similarly made.
After Compline, Zaxo
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