Dot Product

Definition of Dot Product

Given two vectors in $R^2$, define their dot product as:

$$\mathbf{x}\cdot \mathbf{y}=(x_1,y_1)\cdot (x_2,y_2)=x_1{x}_2+y_1{y}_2$$

Given two vectors in $R^n$, define their dot product as:

$$\mathbf{x}\cdot \mathbf{y}=(x_1,x_2,\cdots ,x_n)\cdots (y_1,y_2,\cdots ,y_n)=x_1{y}_1+x_2{y}_2+\cdots +x_n{y}_n$$

One geometric interpretation of the dot product of $\mathbf{u}$ and $\mathbf{v}$ is that it's the component of $\mathbf{u}$ in the direction of $\mathbf{v}$ times the magnitude of $\mathbf{v}$. Thus, one $\mathbf{u}$ and $\mathbf{v}$ are pointing in generally the same direction it's positive, in generally opposite directions it's negative, and when they're perpindicular (orthogonal) it is $0$.

Indeed, the cosine of the angle between two unit vectors is the dot product between them, and in general, the cosine of the angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ is the dot product of the unit vectors in their respective directions:

$$\cos \theta =\frac{\mathbf{u}}{||\mathbf{u}||}\cdot \frac{\mathbf{v}}{||\mathbf{v}||}=\frac{\mathbf{u}\cdot \mathbf{v}}{||\mathbf{u}||||\mathbf{v}||}$$

$$\mathbf{u}\cdot \mathbf{v}=||\mathbf{u}||||\mathbf{v}||\cos \theta$$

Properties of Dot Product

For these properties, let $x,y,z\in {\mathbb{R}}^n$ and $c\in \mathbb{R}.$

$$\mathbf{x}\cdot \mathbf{y}=\mathbf{y}\cdot \mathbf{x}\mathrm{\forall }x,y\in {\mathbb{R}}^n\text{(Commutative Property)}$$

$$\mathbf{x}\cdot \mathbf{x}={||\mathbf{x}||}^2\ge 0,\mathbf{x}\cdot \mathbf{x}=0\text{iff}\mathbf{x}=0\text{(Property of Magnitude)}$$

$$c(\mathbf{x}\cdot \mathbf{y})=(c\mathbf{x})\cdot \mathbf{y}=(c\mathbf{y})\cdot \mathbf{x}\text{(Associative Property)}$$

$$\mathbf{x}\cdot (\mathbf{y}+\mathbf{z})=\mathbf{x}\cdot \mathbf{y}+\mathbf{x}\cdot \mathbf{z}\text{(Distributive Property over Addition)}$$