Dot Product

The dot product can be defined for two vectors and by

(1)

where is the angle between the vectors and is the norm. It follows immediately that if is perpendicular to . The dot product therefore has the geometric interpretation as the length of the projection of onto the unit vector when the two vectors are placed so that their tails coincide.

By writing

			(2)
			(3)

it follows that (1) yields

			(4)
			(5)
			(6)
			(7)

So, in general,

			(8)
			(9)

This can be written very succinctly using Einstein summation notation as

(10)

The dot product is implemented in the Wolfram Language as Dot[a, b], or simply by using a period, a . b.

The dot product is commutative

(11)

and distributive

(12)

The associative property is meaningless for the dot product because is not defined since is a scalar and therefore cannot itself be dotted. However, it does satisfy the property

(13)

for a scalar.

The derivative of a dot product of vectors is

(14)

The dot product is invariant under rotations

			(15)
			(16)
			(17)
			(18)
			(19)
			(20)

where Einstein summation has been used.

The dot product is also called the scalar product and inner product. In the latter context, it is usually written . The dot product is also defined for tensors and by

(21)

So for four-vectors and , it is defined by

			(22)
			(23)
			(24)

where is the usual three-dimensional dot product.