The scalar product of two vectors and is denoted . Because of the notation for the product, with a dot between the two factors, the product is also called dot product.

Although both factors and are vectors, the value is a scalar. This explains the name "scalar product". Mathematicians say that the scalar product of two vectors is a rule that assigns a scalar to two vectors.

The manner in which the scalar product of two vectors is to be evaluated will now be described, in one and more than one dimension and in each of these cases in geometric and analytic fashion. The scalar product in one dimension is described below, in two and more dimensions on Page 2.

Scalar product in one dimension

In this case the two vectors either have the same direction, as in Figure 1 below,

Figure 1

or are oppositely directed, as in Figure 2 below.

Figure 2

(a) Geometric definition

= ab, if and have the same direction.(1)

= -ab, if and have opposite directions.(2)

a and b denote the magnitudes of and .

(b) Analytic definition

= a_xb_x.(3)

a_x and b_x are the x-components of and , respectively, relative to an x-axis that is parallel to the two vectors. One such axis is illustrated in Figure 3 below.

Figure 3

The x-axis in Figure 3 has been chosen to point in the direction of vector . It could also have been chosen to point in the opposite direction, that of vector .

Comment. The signs of the scalar components a_x and b_x depend on the direction of the x-axis, but the product a_xb_x does not. In the case illustrated in Figure 3, a_x = -a and b_x = b. Therefore, a_xb_x = = -ab, as in Eq.(2) above.

If the direction of the x-axis were reversed, the signs of a_x and b_x would be reversed also, but the product a_xb_x and the scalar product would remain unchanged.

The scalar product is a geometric quantity that depends only on the two vectors and and not on the choice of coordinate axis.