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

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