Vector algebra relations

Summary

The following are important identities in vector algebra. Identities that only involve the magnitude of a vector $\|\mathbf A\|$ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier.