The scalar product

Two vectors, labelled a and b, are drawn as line segments. Their start points coincide and their end points do not. There is an acute angle of theta between them. Each line segment has an arrow on it pointing from the start point to the end point.

Figure 1: Two vectors with angle between them.

Definition 1.1 (Scalar product).

The scalar product of $a$ and $b$ is

a \cdot b = | a | | b | cos θ

where

$| a |$ is the modulus of $a$ ,
$| b |$ is the modulus of $b$ , and
$θ$ is the angle between $a$ and $b$ .

Remark 1.2.

It is important to use the dot symbol for the scalar product (also called the dot product). You must not use a $\times$ symbol as this denotes the vector product which is defined differently.

Example 1.3.

Let

a = (\begin{matrix} 2 \\ 2 \end{matrix}) and b = (\begin{matrix} 4 \\ 0 \end{matrix}) .

The angle between these vectors is $θ = 45^{\circ}$ . Then $| a | = \sqrt{8}$ and $| b | = 4$ . So,

\begin{array}{l} a \cdot b = (\begin{matrix} 2 \\ 2 \end{matrix}) \cdot (\begin{matrix} 4 \\ 0 \end{matrix}) & = | a | | b | cos θ \\ = \sqrt{8} \times 4 \times cos 45^{\circ} \\ = 4 \sqrt{8} \times \frac{1}{\sqrt{2}} = 4 \frac{\sqrt{8}}{\sqrt{2}} = 4 \sqrt{4} = 8 . \end{array}

Note that the result is a scalar, not a vector.

1 The scalar product