Vector algebra relations

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