From b1c3adb7c13d1e180a87f3e2f11418863a9d92db Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Mon, 15 Oct 2018 16:49:34 +1100
Subject: [PATCH] vectors - 20a&b

---
 spec/vectors.md | 64 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 64 insertions(+)

diff --git a/spec/vectors.md b/spec/vectors.md
index 4c73237..5c6df9c 100644
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -1,11 +1,75 @@
+---
+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
 
 
 ![](graphics/vectors-intro.png)
 
 ## 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 $\hat{\vec{a}}$
+
-- 
2.49.0