---
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
header-includes:
- - \documentclass{standalone}
- - \usepackage{cleveref}
- - \usepackage{harpoon}
- - \usepackage{accent}
- - \usepackage{amsmath}
-...
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
+
+---
# Vectors
Vectors may describe a position relative to $O$.
-For a point $A$, the position vector is $\boldsymbol{OA}$
+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:
-$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$
+$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
+
+{#id .class width=20%}
+{#id .class width=10%}
## Column vector notation
$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
+## 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. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
+
+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
$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$