Cross Product

Taught in Math 115

Cross Product §

[!Definition 1.5.1]

Let $\vec{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$ and $\vec{y}=\left[\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right]$ be two vectors in ${\mathbb{R}}^3$ The vector

$\vec{x}$ x $\vec{y}=\left[\begin{array}{c}{x}_2{y}_3-y_2{x}_3\\ -(x_1{y}_3-y_1{x}_3)\\ x_1{y}_2-y_1{x}_2\end{array}\right]\in {\mathbb{R}}^3$

is called the cross product (or the vector product) of $\vec{x}$ and $\vec{y}$

Properties of Cross Product §

Let $\vec{x},\vec{y},\vec{w}\in {\mathbb{R}}^3$ and c $\in \mathbb{R}$. Then

1. $\vec{x}$ x $\vec{y}$ $\in {\mathbb{R}}^3$
2. $\vec{x}\times \vec{0}=\vec{0}=\vec{0}\times \vec{x}$
3. $\vec{x}\times \vec{x}=\vec{0}$
4. $\vec{x}\times \vec{y}=-(\vec{y}\times \vec{x})$
5. $(c\vec{x})\times \vec{y}=c(\vec{x}\times \vec{y})=\vec{x}\times (c\vec{y})$
6. $\vec{w}\times (\vec{x}\pm \vec{y})=(\vec{w}\times \vec{x})\pm (\vec{w}\times \vec{y})$
7. $(\vec{x}\pm \vec{y})\times \vec{w}=(\vec{x}\times \vec{w})\pm (\vec{y}\times \vec{w})$

Cross Product is NOT associative

$(\vec{x}\times \vec{y})\times \vec{w}\ne \vec{x}\times (\vec{y}\times \vec{w})$