When placed between two vectors, the dot operator "⋅" signifies that the dot product of the two vectors is to be calculated. This operation is fundamental to matrix multiplication, a computation-intensive task frequently performed in scientific calculations.

The dot product (also known as scalar product or inner product) of two vectors is defined as follows. (Three-dimensional vectors are used for simplicity.) Consider the vectors a = [x_a y_a z_a] and b = [x_b y_b z_b]. Their dot product is given by

• a ⋅ b = x_ax_b + y_ay_b + z_az_b.

Note that the dot operator is only defined for two vectors of equal length (or dimensionality), and that its result is not a vector but a scalar (a regular number).

Matrix multiplications can be carried out as a series of dot product operations. For example, multiplying the 3 x 3 matrix M by the 3-dimensional vector d is accomplished as follows. Say that the matrix M consists of three rows, each row a three dimensional vector:

Since each of the three dot products in the column to the right is a scalar number, the column is itself a three-dimensional vector.

It can also be shown that a ⋅ b = |a| |b| cos , where |a| is the length of vector a, |b| is the length of vector |b| and is the angle between them. It is thus simple to determine cos when the vectors a and b are known, and so to determine itself.