# 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 $x⋅y =∑_{i=1}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+…(1)$ $x⋅y =∥x∥∥y ∥cosθ(2)$

For example, for 2D vectors, $x⋅y =x_{1}y_{1}+x_{2}y_{2}$

In Physics terms we write $A⋅B=ABcosθ$ , where $A$ and $B$ denote the magnitude of the vector.

### Properties of Dot Products

Let $w,x,y ∈R_{n}$ and $c∈R$. $x⋅y ∈R$ $x⋅y =y ⋅x$ $x⋅0=0$ $x⋅x=∥x∥_{2}$ $(cx)⋅y =c(x⋅y )=x⋅(cy )$ $w⋅(x±y )=w⋅x±w⋅y $

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