vectors ref

diff --git a/spec/vectors.md b/spec/vectors.md
index 8fa7208311c7603d8b338f3cfbb4e7dfeed9c728..dcdf2c5d572f3020f9bbaadc2b7beada7d73256c 100644 (file)
--- a/spec/vectors.md
+++ b/spec/vectors.md
@@ -1,18 +1,19 @@
 ---
 ---

-header-includes:
-  - \documentclass{standalone}
-  - \usepackage{cleveref}
-  - \usepackage{harpoon}
-  - \usepackage{accent}
-  - \usepackage{amsmath}
-...
+geometry: margin=2cm
+<!-- columns: 2 -->
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+
+---

 
 # Vectors
 
 - **vector:** a directed line segment  
 - arrow indicates direction
 - length indicates magnitude
 
 # Vectors
 
 - **vector:** a directed line segment  
 - arrow indicates direction
 - length indicates magnitude

-- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
+- notated as $\vec{a}, \widetilde{A}$

 - column notation: $\begin{bmatrix}
        x \\ y
      \end{bmatrix}$
 - column notation: $\begin{bmatrix}
        x \\ y
      \end{bmatrix}$


@@ -87,6 +88,19 @@ Produces a real number, not a vector.
 
 $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 
 
 $$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
 

+## 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 $\boldsymbol{b}$:  
+$\boldsymbol{a \cdot b}=\{
+                \begin{array}{ll}
+                  |\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction} \\
+                  -|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions} \\
+                \end{array}$
+

 ## Geometric scalar products
 
 $$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
 ## Geometric scalar products
 
 $$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$

diff --git a/spec/vectors.pdf b/spec/vectors.pdf
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