Products

Dot Product

Definition

Given 2 nonzero vectors $\vec{U}$ and $\vec{V}$ in 2D or 3D, their dot product is $\vec{U}\cdot\vec{V}= |\vec{U}||\vec{V}|\cos{\theta}$ where $\theta$ is the angle between $\vec{U}$ and $\vec{V}$ with $0\leq\theta\leq\pi$

Equation

$\vec{U}=\langle U_1,U_2,U_3\rangle$

$\vec{V}=\langle V_1,V_2,V_3\rangle$

$U \cdot V = U_1V_1+U_2V_2+U_3V_3=|\vec{U}||\vec{V}|\cos{\theta}$

Key Notes

Dot product is commutative

Orthogonal Vectors

Two vectors are orthogonal if $\vec{U}\cdot\vec{V}=0$ . In 2D and 3D spaces, orthogonal means perpendicular ( $\frac{\pi}{2}$ angle between them)

Key Notes

There are an infinite number of orthogonal vectors to any one vector when length is not specified

Projections

The orthogonal projection is a vector that goes along $\vec{U}$ with length $|\vec{V}|\cos{\theta}$

Equation

$proj_{\vec{U}}\vec{V}= \frac{\vec{U}}{|\vec{U}|}|\vec{V}|\cos{\theta}$

Cross Product

Definition

Given two nonzero vectors $u$ and $v$ in ${\rm I\!R}^3$ , the cross product is defined as $\vec{u} \times \vec{v} = |\vec{u}||\vec{u}|\sin{\theta}\vec{n}$ where $0\leq\theta\leq\pi$ is the angle between $\vec{u}$ and $\vec{v}$ and $\vec{n}$ is the unit vector which is normal (perpendicular) to $\vec{u}$ and $\vec{v}$ whose direction is determined by the right hand rule

Equation $\vec{u} \times \vec{v} = |\vec{u}||\vec{u}|\sin{\theta}\vec{n}$

Key Notes

$\vec{u}$ and $\vec{v}$ are parallel if and only if $\vec{u}\times\vec{v}=0$
If $\vec{u}$ and $\vec{v}$ are two sides of a parallelogram, then the area of the paralllelogram is $|\vec{u}\times\vec{v}| = |\vec{u}||\vec{v}|\sin{\theta}$
$\vec{v}\times\vec{u}=-(\vec{u}\times\vec{v})$
$\vec{u}\times\vec{v}$ is orthogonal to $\vec{u}$ and $\vec{v}$ which means that $(\vec{u}\times\vec{v})\cdot\vec{u} = 0$ and $(\vec{u}\times\vec{v})\cdot\vec{v} = 0$