+---
+header-includes:
+ - \documentclass{standalone}
+ - \usepackage{cleveref}
+ - \usepackage{harpoon}
+ - \usepackage{accent} \newcommand{\vect}[1]{\accentset{\rightharpoonup}{#1}}
+---
+
# Vectors
- **vector:** a directed line segment
- arrow indicates direction
- length indicates magnitude
- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
+- column notation: $\begin{bmatrix}
+ x \\ y
+ \end{bmatrix}$
+- vectors with equal magnitude and direction are equivalent
## Vector addition
+
+$\vec{u} + \vec{v}$ can be represented by drawing each vector head to tail then joining the lines.
+Addition is commutative (parallelogram)
+
+## Scalar multiplication
+
+For $k \in \mathbb{R}^+$, $k\vec{u}$ has the same direction as $\vec{u}$ but length is multiplied by a factor of $k$.
+
+When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
+
+## Vector subtraction
+
+To find $\vec{u} - \vec{v}$, add $\vec{-v}$ to $\vec{u}$
+
+## Parallel vectors
+
+Parallel vectors have same direction or opposite direction.
+
+**Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\vec{u} = k \vec{v}$**
+
+## Position vectors
+
+Vectors may describe a position relative to $O$.
+
+For a point $A$, the position vector is $\vec{OA}$
+
+## Linear combinations of non-parallel vectors
+
+If two non-zero vectors $\vec{a}$ and $\vec{b}$ are not parallel, then:
+
+$$m\vec{a} + n\vec{b} = p \vec{a} + q \vec{b}\quad\text{implies}\quad m = p, \> n = q$$
+
+## Column vector notation
+
+A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}$
+
+## Component notation
+
+A vector $\vec{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\vec{u} = x\vec{i} + y\vec{j}$.
+$\vec{u}$ is the sum of two components $x\vec{i}$ and $y\vec{j}$
+Magnitude of vector $\vec{u} = x\vec{i} + y\vec{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
+
+Basic algebra applies:
+$(x\vec{i} + y\vec{j}) + (m\vec{i} + n\vec{j}) = (x + m)\vec{i} + (y+n)\vec{j}$
+Two vectors equal if and only if their components are equal.
+
+## Unit vectors
+
+A vector of length 1. $\vec{i}$ and $\vec{j}$ are unit vectors.
+
+A unit vector in direction of $\vec{a}$ is denoted by $\hat{\vec{a}}$
+
+Also, unit vector of $\vec{a}$ can be defined by $\vec{a} \cdot {|\vec{a}|}$
+
+## Scalar products / dot products
+
+If $\vec{a} = a_i \vec{i} + a_2 \vec{j}$ and $\vec{b} = b_i \vec{i} + b_2 \vec{j}$, the dot product is:
+$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$
+
+Produces a real number, not a vector.
+
+$$\vec{a} \cdot \vec{a} = |\vec{a}|^2$$
+
+## Geometric scalar products
+
+$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$
+
+where $0 \le \theta \le \pi$
+
+## Perpendicular vectors
+
+If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$ (since $\cos 90 = 0$)
+
+## Finding angle between vectors
+
+$$\cos \theta = {{\vec{a} \cdot \vec{b}} \over {|\vec{a}| |\vec{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\vec{a}| |\vec{b}|}}$$
+
+
+## Vector projections
+