AHL 3.13—Scalar and vector products

Scalar Product (Dot Product): For two vectors $\mathbf{a}=\left(\begin{array}{c}{a}_1{\text{a}}_2{\text{a}}_3\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}{b}_1{\text{b}}_2{\text{b}}_3\end{array}\right)$ in three-dimensional space, their scalar product is defined as:
$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$
The result is always a scalar (a real number), not a vector.

The dot between the two vectors ($\cdot$) is the standard notation. Do not confuse this with ordinary multiplication , this is a specific operation between two vectors.

The scalar product obeys a number of algebraic properties that mirror those of ordinary multiplication, with one important addition:

1. Commutativity: $\mathbf{v}\cdot \mathbf{w}=\mathbf{w}\cdot \mathbf{v}$

2. Distributivity over addition: $\mathbf{u}\cdot (\mathbf{v}+\mathbf{w})=\mathbf{u}\cdot \mathbf{v}+\mathbf{u}\cdot \mathbf{w}$

3. Scalar multiplication: $(k\mathbf{v})\cdot \mathbf{w}=k(\mathbf{v}\cdot \mathbf{w})$ for any scalar $k$

4. Self-dot product: $\mathbf{v}\cdot \mathbf{v}=|\mathbf{v}{|}^2$

Property 4 is particularly useful. Since $\mathbf{v}\cdot \mathbf{v}=v_1^2+v_2^2+v_3^2$ and $|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}$, it follows directly that:
$\mathbf{v}\cdot \mathbf{v}=|\mathbf{v}{|}^2$