Length

$||\vec{a}||=\sqrt{a_1^2+\dots+a_n^2}=\sqrt{\vec{a}\cdot\vec{a}}$

$||\vec{a}||^2=\vec{a}\cdot\vec{a}$

Dot Product

(Aka. "Scalar Product")

Measures how much one vector points in the direction of another.

Let $\vec{a}=\begin{bmatrix} a_1 \\ \vdots \\ a_n \end{bmatrix},b=\begin{bmatrix} b_1 \\ \vdots \\ b_n \end{bmatrix}$

Geometric Definition

$\vec{a}\cdot\vec{b}=||\vec{a}||\cdot||\vec{b}||\cdot \cosθ$

Algebraic Definition

$\vec{a}\cdot\vec{b}=a_1b_1+\dots+a_nb_n$

Vector Relationships

Orthogonal / Perpendicular

Vectors $\vec{u}$ and $\vec{v}$ are orthogonal to each other if $\vec{u}\cdot\vec{v}=0$

• One of the vectors' lengths is zero $\vec{0}$
• $\cos\theta=0, \theta=90\degree$

Direction

Direction: $\vec{u}$ points in the direction of $\vec{v}$ if $\vec{u}=k\vec{v}$ for some $k$.

Positive Direction: $\vec{u}$ points in the positive direction of $\vec{v}$ if $\vec{u}=k\vec{v}$ for some $k>0$.

$\cos\theta=\frac{\vec{p}\cdot\vec{q}}{||\vec{p}||||\vec{q}||}$