Cross Product

$$\ast \vec{a}\times \vec{b}=\left(|a||b|\sin (\theta )\right)\vec{n}=\vec{c}\ast$$

where $\vec{n}$ is the unit normal vector of the plane described by $\vec{a}$ and $\vec{b}$ Direction of $\vec{c}$ according to right hand rule

Properties

$$\begin{array}{rl}\vec{a}\times \vec{b}& =-(\vec{b}\times \vec{a})\\ \vec{a}\times (\vec{b}+\vec{c})& =\vec{a}\times \vec{b}+\vec{a}\times \vec{c}\\ (\vec{a}+\vec{b})\times \vec{c}& =\vec{a}\times \vec{c}+\vec{b}\times \vec{c}\\ (C\vec{a})\times \vec{b}=\vec{a}\times (C\vec{b})& =C(\vec{a}\times \vec{b})\\ \vec{a}\times (\vec{b}\times \vec{c})& =\vec{b}\times (\vec{c}\times \vec{a})=\vec{c}\times (\vec{a}\times \vec{b})\end{array}$$

Magnitude

$$\underset{\text{Area}({▰}_{a,b})}{\underbrace{|\vec{a}\times \vec{b}|=|a||b|\sin (\theta )}}⟹\vec{a}\parallel \vec{b}⟺\vec{a}\times \vec{b}=\vec{0}$$