Dot Product

This was initially learned in linear algebra, but has a lot of applications in engineering. It is used in projections.

Why did the dot product come from? Go to the bottom of the page for a proof.

We have two definitions of the dot product $\vec{x}\cdot \vec{y}={\sum }_{i=1}^n{x}_i{y}_i=x_1{y}_1+x_2{y}_2+\dots (1)$ $\vec{x}\cdot \vec{y}=\Vert \vec{x}\Vert \Vert \vec{y}\Vert \cos \theta (2)$

For example, for 2D vectors, $\vec{x}\cdot \vec{y}=x_1{y}_1+x_2{y}_2$

In Physics terms we write $\vec{A}\cdot \vec{B}=AB\cos \theta$ , where $A$ and $B$ denote the magnitude of the vector.

Properties of Dot Products

Let $\vec{w},\vec{x},\vec{y}\in {\mathbb{R}}^n$ and $c\in \mathbb{R}$. $\vec{x}\cdot \vec{y}\in \mathbb{R}$ $\vec{x}\cdot \vec{y}=\vec{y}\cdot \vec{x}$ $\vec{x}\cdot \vec{0}=0$ $\vec{x}\cdot \vec{x}=\Vert x{\Vert }^2$ $(c\vec{x})\cdot \vec{y}=c(\vec{x}\cdot \vec{y})=\vec{x}\cdot (c\vec{y})$ $\vec{w}\cdot (\vec{x}\pm \vec{y})=\vec{w}\cdot \vec{x}\pm \vec{w}\cdot \vec{y}$

• dot product > 0 $⟹$ angle is acute
• dot product < 0 $⟹$ angle is obtuse
• dot product = 0 $⟹$ two vectors are orthogonal

Proof of Dot Product