spec / vectors.mdon commit dot products and vector angles (a97a120)
   1---
   2geometry: margin=2cm
   3<!-- columns: 2 -->
   4graphics: yes
   5tables: yes
   6author: Andrew Lorimer
   7classoption: twocolumn
   8header-includes:
   9- \usepackage{harpoon}
  10- \usepackage{amsmath}
  11- \pagenumbering{gobble}
  12
  13---
  14
  15# Vectors
  16
  17- **vector:** a directed line segment  
  18- arrow indicates direction
  19- length indicates magnitude
  20- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
  21- column notation: $\begin{bmatrix}
  22       x \\ y
  23     \end{bmatrix}$
  24- vectors with equal magnitude and direction are equivalent
  25
  26
  27![](graphics/vectors-intro.png)
  28
  29## Vector addition
  30
  31$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.  
  32Addition is commutative (parallelogram)
  33
  34## Scalar multiplication
  35
  36For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$.
  37
  38When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
  39
  40## Vector subtraction
  41
  42To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$
  43
  44## Parallel vectors
  45
  46Parallel vectors have same direction or opposite direction.
  47
  48**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**
  49
  50## Position vectors
  51
  52Vectors may describe a position relative to $O$.
  53
  54For a point $A$, the position vector is $\overrightharp{OA}$
  55
  56\vfill\eject
  57
  58## Linear combinations of non-parallel vectors
  59
  60If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:
  61
  62$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$
  63
  64![](graphics/parallelogram-vectors.jpg){#id .class width=20%}
  65![](graphics/vector-subtraction.jpg){#id .class width=10%}
  66
  67## Column vector notation
  68
  69A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}$
  70
  71## Component notation
  72
  73A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.  
  74$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$  
  75Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
  76
  77Basic algebra applies:  
  78$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  
  79Two vectors equal if and only if their components are equal.
  80
  81## Unit vectors
  82
  83A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
  84
  85A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:
  86
  87$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$
  88
  89Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$
  90
  91## Scalar products / dot products
  92
  93If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is:
  94$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
  95
  96Produces a real number, not a vector.
  97
  98$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
  99
 100**on CAS:** `dotP([a b c], [d e f])`
 101
 102## Scalar product properties
 103
 1041. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
 1052. $\boldsymbol{a \cdot 0}=0$
 1063. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
 1074. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
 1085. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
 1096. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
 110
 111For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
 112$$\boldsymbol{a \cdot b}=\begin{cases}
 113|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 114-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 115\end{cases}$$
 116
 117## Geometric scalar products
 118
 119$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
 120
 121where $0 \le \theta \le \pi$
 122
 123## Perpendicular vectors
 124
 125If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)
 126
 127## Finding angle between vectors
 128
 129**positive direction**
 130
 131$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
 132
 133**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
 134
 135
 136## Vector projections
 137
 138Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.
 139
 140$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 141
 142## Vector proofs
 143
 144**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
 145**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 \lt \mu \lt 1$)
 146
 147Useful vector properties:
 148
 149- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
 150- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
 151- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
 152- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
 153
 154## Linear dependence
 155
 156Vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent if they are non-parallel and:
 157
 158$$k\vec{a}+l\vec{b}+m\vec{c} = 0$$
 159$$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$
 160
 161$\vec{a}, \vec{b},$ 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.
 162
 163Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$
 164
 165## Three-dimensional vectors
 166
 167Right-hand rule for axes - $z$ is up or out of page.
 168
 169## Angle between vector and axis
 170
 171Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}, \vec{k}$ directions.
 172
 173For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
 174$$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$
 175
 176**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\vec{i} + b\vec{j} + c\vec{k}$ and $x$-axis
 177