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

diff --git a/spec/vectors.md b/spec/vectors.md
index a2122b07af967d273b84716b4a25025dcb83a087..2374b2a845082ae10b1bc91e64671e8f6a0afd27 100644 (file)
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -1,11 +1,16 @@
 ---
+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
 
@@ -46,13 +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
 
@@ -87,6 +97,18 @@ Produces a real number, not a vector.
 
 $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 
+## 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
 
 $$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
@@ -108,5 +130,18 @@ Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitud
 
 $$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 
+## Vector proofs
+
+**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
+**Collinear points -** $\ge$ 3 points lie on the same line
+
+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 $\boldsymbol{a} \cdot \boldsymbol{b}=0$
+- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
+
+