Vector

A vector has a magnitude and a direction. Do not get it confused with a scalar.

The vector magnitude is always positive.

Use in Physics

Cartesian Vector Notation

This notation is based on the concept of unit vectors. $\vec{F}=F_x\hat{i}+F_y\hat{j}+F_z\hat{k}$

Unit Vector → ${\hat{e}}_{AB}$

Norm/Magnitude

The norm of $\vec{x}\in {\mathbb{R}}^n$ is $\Vert \vec{x}\Vert =\sqrt{x_1^2+\cdots +x_n^2}$

Properties of Norms. Let $\vec{x},\vec{y}\in {\mathbb{R}}^n$ and $c\in \mathbb{R}$. Then $\Vert \vec{x}\Vert \ge 0\text{with equality}⟺\vec{x}=\vec{0}$ $\Vert c\vec{x}\Vert =|c|\Vert \vec{x}\Vert$ $\Vert \vec{x}+\vec{y}\Vert \le \Vert \vec{x}\Vert +\Vert \vec{y}\Vert \text{which is known as the Triangle Inequality}$

Dot Product

$\vec{x}\cdot \vec{y}=\Vert \vec{x}\Vert \Vert \vec{y}\Vert \cos \theta$

A dot product result is just a scalar, but it still has units. This unit is the product of the units of the two vectors.

This is used in many applications to determine the projections of vectors onto various directions.

Angle between two vectors

From the dot product formula, we can derive the angle. $\theta ={\cos }^{-1}(\frac{\vec{x}\cdot \vec{y}}{\Vert x\Vert \Vert y\Vert })$

Cross Product

The cross product of two vectors is also a vector. Its unit is the product of the units of the two original vectors. This is used to calculate angular momentum, torque, and magnetic force.

Complex Vectors

For $z_1,\dots ,z_n\in \mathbb{C}$, we define $\vec{z}=\left[\begin{array}{c}{z}_1\\ ⋮\\ z_n\end{array}\right]$ $\overline{\stackrel{⃗}{z}}=\left[\begin{array}{c}\overline{z_1}\\ ⋮\\ \overline{z_n}\end{array}\right]$

$⟨\vec{z},\vec{w}⟩=\overline{\vec{z}}\cdot \vec{w}$

(Properties of Complex Inner Products). Let $\vec{v},\vec{w},\vec{z}\in {\mathbb{C}}^n$ and $\alpha \in \mathbb{C}$. Then

$⟨\vec{z},\vec{z}⟩\ge 0\text{with equality}⟺\vec{z}=\vec{0}$ $⟨\vec{z},\vec{w}⟩=\overline{⟨\vec{w},\vec{z}⟩}$ $⟨\vec{v}+\vec{w},\vec{z}⟩=⟨\vec{v},\vec{z}⟩+⟨\vec{w},\vec{z}⟩$ $⟨\alpha \vec{z},\vec{w}⟩=\bar{\alpha }⟨\vec{z},\vec{w}⟩\text{and}⟨\vec{z},\alpha \vec{w}⟩=\alpha ⟨\vec{z},\vec{w}⟩$ $|⟨\vec{z},\vec{w}⟩|\le \Vert z\Vert \Vert w\Vert \text{Cauchy-Shwarz Inequality}$ $\Vert z+w\Vert \le \Vert z\Vert +\Vert w\Vert \text{Triangle Inequality}$

Unit Vector

Definition: A vector $\vec{x}\in {\mathbb{R}}^n$ is a unit vector if $\Vert \vec{x}\Vert =1$. $\hat{e}=\frac{\vec{x}}{\Vert \vec{x}\Vert }$

In physics, we use the ${\hat{e}}_{AB}$ notation to denote the unit vector.