From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 6 Nov 2018 10:08:20 +0000 (+1100)
Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes
X-Git-Tag: yr11~5
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/62daddfa5acf936a2fc7cbec75456f083dd1e32a?hp=69d695fd3eea1190d9f2e8002c25b92ea0da042f

Merge branch 'master' of ssh://charles/tank/andrew/school/notes
---

diff --git a/physics/final.pdf b/physics/final.pdf
index 331a41e..375bdb1 100644
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diff --git a/physics/final.tex b/physics/final.tex
index 57f04de..fabacef 100644
--- a/physics/final.tex
+++ b/physics/final.tex
@@ -227,6 +227,7 @@
       \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}
@@ -318,7 +319,7 @@
   \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*}
@@ -437,7 +438,7 @@
   \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 $\theta_r$
 
diff --git a/spec/graphics/parallelogram-vectors.jpg b/spec/graphics/parallelogram-vectors.jpg
new file mode 100755
index 0000000..8bafae1
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diff --git a/spec/graphics/vector-subtraction.jpg b/spec/graphics/vector-subtraction.jpg
new file mode 100755
index 0000000..f94832b
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diff --git a/spec/vectors.md b/spec/vectors.md
index dcdf2c5..2374b2a 100644
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -5,6 +5,10 @@ graphics: yes
 tables: yes
 author: Andrew Lorimer
 classoption: twocolumn
+header-includes:
+- \usepackage{harpoon}
+- \usepackage{amsmath}
+- \pagenumbering{gobble}
 
 ---
 
@@ -13,7 +17,7 @@ classoption: twocolumn
 - **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: $\begin{bmatrix}
        x \\ y
      \end{bmatrix}$
@@ -47,13 +51,18 @@ Parallel vectors have same direction or opposite direction.
 
 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:
 
-$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$
+$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
+
+![](graphics/parallelogram-vectors.jpg){#id .class width=20%}
+![](graphics/vector-subtraction.jpg){#id .class width=10%}
 
 ## Column vector notation
 
@@ -95,11 +104,10 @@ $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 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
 
diff --git a/spec/vectors.pdf b/spec/vectors.pdf
index b60fffc..c468f7f 100644
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