+---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+header-includes:
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
+
+---
+
# 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
+
+$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.
+Addition is commutative (parallelogram)
+
+## Scalar multiplication
+
+For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ 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 $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$
+
+## Parallel vectors
+
+Parallel vectors have same direction or opposite direction.
+
+**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**
+
+## Position vectors
+
+Vectors may describe a position relative to $O$.
+
+For a point $A$, the position vector is $\overrightharp{OA}$
+
+\vfill\eject
+
+## Linear combinations of non-parallel vectors
+
+If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:
+
+$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
+
+{#id .class width=20%}
+{#id .class width=10%}
+
+## 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 $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.
+$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$
+Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
+
+Basic algebra applies:
+$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$
+Two vectors equal if and only if their components are equal.
+
+## Unit vectors
+
+A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
+
+A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:
+
+$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$
+
+Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$
+
+## Scalar products / dot products
+
+If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is:
+$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
+
+Produces a real number, not a vector.
+
+$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
+
+**on CAS:** `dotP([a b c], [d e f])`
+
+## Scalar product properties
+
+1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
+2. $\boldsymbol{a \cdot 0}=0$
+3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
+4. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
+5. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
+6. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
+
+For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:
+$$\boldsymbol{a \cdot b}=\begin{cases}
+|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
+-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
+\end{cases}$$
+
+## Geometric scalar products
+
+$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
+
+where $0 \le \theta \le \pi$
+
+## Perpendicular vectors
+
+If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)
+
+## Finding angle between vectors
+
+**positive direction**
+
+$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
+
+**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
+
+
+## Vector projections
+
+Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.
+
+$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
+
+Scalar resolute of $\vec{a}$ on $\vec{b} = |\vec{u}| = \vec{a} \cdot \hat{\vec{b}}$ (results in a scalar)
+Vector resolute of $\vec{a}$ perpendicular to $b$ is equal to $\vec{a} - \vec{u}$ where $\vec{u}$ is vector projection of $\vec{a}$ on $\vec{b}$
+
+## Vector proofs
+
+**Concurrent lines -** $\ge$ 3 lines intersect at a single point
+**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$)
+
+Useful vector properties:
+
+- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
+- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
+- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
+- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
+
+## Linear dependence
+
+Vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent if they are non-parallel and:
+
+$$k\vec{a}+l\vec{b}+m\vec{c} = 0$$
+$$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$
+
+$\vec{a}, \vec{b},$ and $\vec{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.
+
+Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$
+
+## Three-dimensional vectors
+
+Right-hand rule for axes - $z$ is up or out of page.
+
+## Angle between vector and axis
+
+Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}, \vec{k}$ directions.
+
+For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
+$$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$
+
+**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\vec{i} + b\vec{j} + c\vec{k}$ and $x$-axis
+
+## Collinearity
+
+Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$