Angle between two 3D vectors

This is something I noticed the other day. Someone had posted a method for finding the angle between two vectors in three dimensions, using the dot product and inverse cosine. But there is better approach, i.e. one that is generally more efficient, is certainly more accurate for some vectors, and can be more informative.

Angle between two vectors

Another use for inverse trigonometric functions is finding the angle between two vectors. Again there is more than one way to do it, and again the most straightforward approach is often not the best.

Inverse trigonometry

The functions asin, acos and atan are together the three principal inverse trigonometric functions; that is they are the inverse of the sine, cosine and tangent functions, the commonest trigonometric functions that describe all the ratios of the sides of a right-angled triangle.