Dot Product

The dot product is an operation that takes two vectors of the same dimension and returns a single real number (a scalar), often written with a centered dot like (\mathbf{a} \cdot \mathbf{b}).

Algebraic definition

For vectors in ${\mathbb{R}}^n$

$$\mathbf{a}=\left(\begin{array}{c}{a}_1\\ a_2\\ ⋮\\ a_n\end{array}\right),\mathbf{b}=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right),$$

their dot product is

$$\mathbf{a}\cdot \mathbf{b}=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n.$$

Example in ${\mathbb{R}}^3$

$$\left(\begin{array}{c}1\\ 3\\ -5\end{array}\right)\cdot \left(\begin{array}{c}4\\ -2\\ -1\end{array}\right)1\cdot 4+3\cdot (-2)+(-5)\cdot (-1)4-6+5=3.$$

Geometric definition

If $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^n$

and $\theta$

is the angle between them, then

$\mathbf{a}\cdot \mathbf{b}$

$\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta ,$

where $\Vert \mathbf{a}\Vert$

is the Euclidean length (norm) of $\mathbf{a}$

From this, you also get

$$\mathbf{a}\cdot \mathbf{a}=\Vert \mathbf{a}{\Vert }^2,\Vert \mathbf{a}\Vert =\sqrt{\mathbf{a}\cdot \mathbf{a}}.$$

Basic calculation rules

Let $\mathbf{a},\mathbf{b},\mathbf{c}\in {\mathbb{R}}^n$

and $\lambda \in \mathbb{R}$

Then:

• Commutativity:

  $$\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a}.$$

• Distributivity over addition:

  $$\mathbf{a}\cdot (\mathbf{b}+\mathbf{c})\mathbf{a}\cdot \mathbf{b}+\mathbf{a}\cdot \mathbf{c}.$$

• Homogeneity (scalar multiplication in one slot):

  $$(\lambda \mathbf{a})\cdot \mathbf{b}\lambda (\mathbf{a}\cdot \mathbf{b}),\mathbf{a}\cdot (\lambda \mathbf{b})\lambda (\mathbf{a}\cdot \mathbf{b}).$$

• Positivity:

  $$\mathbf{a}\cdot \mathbf{a}\ge 0\text{and}\mathbf{a}\cdot \mathbf{a}=0⟺\mathbf{a}=0.$$

Worked examples

1. Simple 2D example

Let

$$\mathbf{u}=\left(\begin{array}{c}2\\ -1\end{array}\right),\mathbf{v}=\left(\begin{array}{c}3\\ 4\end{array}\right).$$

Then

$$2\cdot 3+(-1)\cdot 46-42.$$

2. 4D example

Let

$$\mathbf{x}=\left(\begin{array}{c}2\\ 0\\ -3\\ 1\end{array}\right),\mathbf{y}=\left(\begin{array}{c}-1\\ 3\\ 1\\ 2\end{array}\right).$$

Then

$$\mathbf{x}\cdot \mathbf{y}2(-1)+0(3)+(-3)(1)+1(2)-2+0-3+2-3.$$

3. Using the geometric form to find an angle

Let

$$\mathbf{a}=\left(\begin{array}{c}1\\ 2\end{array}\right),\mathbf{b}=\left(\begin{array}{c}2\\ 1\end{array}\right).$$

Compute

$$\mathbf{a}\cdot \mathbf{b}1\cdot 2+2\cdot 14,$$ $$\Vert \mathbf{a}\Vert \sqrt{1^2+2^2}\sqrt{5},\Vert \mathbf{b}\Vert \sqrt{2^2+1^2}\sqrt{5}.$$

So

$$\cos \theta \frac{\mathbf{a}\cdot \mathbf{b}}{\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert }\frac{4}{\sqrt{5}\sqrt{5}}\frac{4}{5},$$

and hence

$$\theta =\arccos \left(\frac{4}{5}\right).$$