linear dependence of vectors (spec)

diff --git a/spec/vectors.md b/spec/vectors.md
index dcdf2c5d572f3020f9bbaadc2b7beada7d73256c..6218b24ed369c915ef0a331990c177d91e295cc7 100644 (file)
--- 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
 
@@ -134,6 +142,13 @@ Useful vector properties:
 - Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
 - $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
 
+## Linear dependence
+
+Vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent if they are non-parallel and:
 
+$$k\vec{a}+l\vec{b}+m\vec{c} = 0$$
+$$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$
 
+$\vec{a}, \vec{b},$ and $\vec{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.
 
+Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$