## Dot Product

The dot product can be defined by

 (1)

where is the angle between the vectors. It follows immediately that if is Perpendicular to . The dot product is also called the Inner Product and written . By writing
 (2) (3)

it follows that (1) yields
 (4)

So, in general,
 (5)

The dot product is Commutative
 (6)

Associative
 (7)

and Distributive
 (8)

The Derivative of a dot product of Vectors is
 (9)

The dot product is invariant under rotations
 (10)

where Einstein Summation has been used.

The dot product is also defined for Tensors and by

 (11)

Arfken, G. Scalar or Dot Product.'' §1.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 13-18, 1985.