\item closer field lines means larger force
\item dot: out of page, cross: into page
\item +ve corresponds to N pole
+ \item Inv. sq. ${E_1 \over E_2} = ({r_2 \over r_1})^2$
\end{itemize}
\includegraphics[height=2cm]{graphics/field-lines.png}
\subsection*{Power transmission}
% \begin{align*}
- \[V_{\operatorname{rms}}={V_{\operatorname{p\rightarrow p}}\over \sqrt{2}} \]
+ \[V_{\operatorname{rms}}={V_{\operatorname{p}}\over \sqrt{2}}={V_{\operatorname{p\rightarrow p}}\over {2 \sqrt{2}}} \]
\[P_{\operatorname{loss}} = \Delta V I = I^2 R = {{\Delta V^2} \over R} \]
\[V_{\operatorname{loss}}=IR \]
% \end{align*}
\subsection*{Refraction}
\includegraphics[height=3.5cm]{graphics/refraction.png}
- When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}
+ When a medium changes character, light is \emph{reflected}, \emph{absorbed}, and \emph{transmitted}. $\lambda$ changes, not $f$.
angle of incidence $\theta_i =$ angle of reflection $\theta_r$
tables: yes
author: Andrew Lorimer
classoption: twocolumn
+header-includes:
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
---
- **vector:** a directed line segment
- arrow indicates direction
- length indicates magnitude
-- notated as $\vec{a}, \widetilde{A}$
+- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
- column notation: $\begin{bmatrix}
x \\ y
\end{bmatrix}$
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
3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:
-$\boldsymbol{a \cdot b}=\{
- \begin{array}{ll}
- |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\
- -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\
- \end{array}$
+$$\boldsymbol{a \cdot b}=\begin{cases}
+|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
+-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
+\end{cases}$$
## Geometric scalar products
