Dot Product

Definitions

• $\vec{a}\cdot \vec{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$
• $\vec{a}\cdot \vec{b}=\frac{1}{2}\left({\Vert \vec{a}\Vert }^2+{\Vert \vec{b}\Vert }^2-{\Vert \vec{b}-\vec{a}\Vert }^2\right)$

Application

$$\vec{a}\cdot \vec{b}=\Vert \vec{a}\Vert \Vert \vec{b}\Vert \cos \theta \text{ for }\theta \in [0,\pi ]$$

Gives an algebraic property that can determine the angle between two oriented segments

$$\vec{a}\cdot \vec{b}=0⟺\vec{a}\perp \vec{b}$$

Algebraic Properties

• $\vec{a}\cdot \left(\vec{b}+\vec{c}\right)=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$
• $\left(s\vec{a}\right)\cdot \vec{b}=s\left(\vec{a}\cdot \vec{b}\right)$
• $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$
• $\vec{a}\cdot \vec{a}=a_1^2+a_2^2+a_3^2=\Vert \vec{a}{\Vert }^2$

Interesting Thing from 3B1B

\vec{a} \cdot \vec{b} = \left\| proj_\vec{a}\vec{b}\right\| \left\|\vec{a}\right\|