Dot Product

The dot product of two nonzero real vectors and is the scalar

(1)

where is the vector angle and is the Euclidean norm. The dot product is zero if either vector is zero or if the two nonzero vectors are perpendicular.

For , let be the unit vector in its direction. The scalar projection of onto this direction is

(2)

For nonzero , this equals . Thus, is the scalar projection multiplied by , not the projection length alone. The vector projection, shown in red in the illustration, is

(3)

and its length is . The scalar projection is negative when is an obtuse angle, whereas the length is always nonnegative. For example, the projection of onto is and has length 4, but . Interchanging the vectors changes the projection but leaves the dot product unchanged.

By writing

			(4)
			(5)

it follows that (1) yields

			(6)
			(7)
			(8)
			(9)

So, in general,

			(10)
			(11)

This can be written very succinctly using Einstein summation notation as

(12)

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

(13)

and distributive

(14)

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

(15)

for a scalar.

The derivative of a dot product of vectors is

(16)

The dot product is invariant under rotations

			(17)
			(18)
			(19)
			(20)
			(21)
			(22)

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

(23)

So for four-vectors and , it is defined by

			(24)
			(25)
			(26)

where is the usual three-dimensional dot product.