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.

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

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