Cross Product

Learned this in linear, but useful in engineering as well.

$|\vec{x}\times \vec{y}|=|\vec{x}||\vec{y}|\sin \theta$

Area of Parallelogram

We have $|\vec{x}\times \vec{y}|=|\vec{x}||\vec{y}|\sin \theta$

Volume of Parallelepiped

A parallelepiped is the three dimensional analogue of a parallelogram.

$V=|\vec{w}\cdot (\vec{x}\times \vec{y})|$

Triple Product

Learned from Cyrill Stachniss during the Fundamental Matrix lecture.

Properties of Cross Products

Properties of Cross Products. Let $\vec{x},\vec{y},\vec{w}\in {\mathbb{R}}^3,c\in \mathbb{R}$. Then $\vec{x}\times \vec{y}\in {\mathbb{R}}^3$ $\vec{x}\times \vec{y}\text{is orthogonal to both}\vec{x}\text{and}\vec{y}$ $\vec{x}\times \vec{0}=\vec{0}\times \vec{x}=\vec{0}$ $\vec{x}\times \vec{x}=\vec{0}$ $\vec{x}\times \vec{y}=-(\vec{y}\times \vec{x})$ $(c\vec{x})\times \vec{y}=c(\vec{x}\times \vec{y})=\vec{x}\times (c\vec{y})$ $\vec{w}\times (\vec{x}\pm \vec{y})=(\vec{w}\times \vec{x})\pm (\vec{w}\times \vec{y})$ $(\vec{x}\pm \vec{y})\times \vec{w}=(\vec{x}\times \vec{w})\pm (\vec{y}\times \vec{w})$

Calculating the Cross Product