spec / vectors.mdon commit [spec] start vector calculus (42fe40b)
   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){#id .class width=20%} 
  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
  46Same or opposite direction
  47
  48$$\boldsymbol{u} || \boldsymbol{v} \iff \boldsymbol{u} = k \boldsymbol{v} \text{ where } k \in \mathbb{R} \setminus \{0\}$$ 
  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 vector $|\hat{\boldsymbol{a}}|=1$
  82
  83\begin{equation}\begin{split}\hat{\boldsymbol{a}} & = {1 \over {|\boldsymbol{a}|}}\boldsymbol{a} \\ & = \boldsymbol{a} \cdot {|\boldsymbol{a}|}\end{split}\end{equation}
  84
  85## Scalar/dot product $\boldsymbol{a} \cdot \boldsymbol{b}$
  86
  87$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
  88
  89**on CAS:** `dotP([a b c], [d e f])`
  90
  91## Scalar product properties
  92
  931. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
  942. $\boldsymbol{a \cdot 0}=0$
  953. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
  964. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
  975. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
  986. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$
  99
 100For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
 101$$\boldsymbol{a \cdot b}=\begin{cases}
 102|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
 103-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
 104\end{cases}$$
 105
 106## Geometric scalar products
 107
 108$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
 109
 110where $0 \le \theta \le \pi$
 111
 112## Perpendicular vectors
 113
 114If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)
 115
 116## Finding angle between vectors
 117
 118**positive direction**
 119
 120$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
 121
 122**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)
 123
 124## Angle between vector and axis
 125
 126Direction of a vector can be given by the angles it makes with $\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}$ directions.
 127
 128For $\boldsymbol{a} = a_1 \boldsymbol{i} + a_2 \boldsymbol{j} + a_3 \boldsymbol{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
 129$$\cos \alpha = {a_1 \over |\boldsymbol{a}|}, \quad \cos \beta = {a_2 \over |\boldsymbol{a}|}, \quad \cos \gamma = {a_3 \over |\boldsymbol{a}|}$$
 130
 131**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ and $x$-axis
 132
 133## Vector projections
 134
 135Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$:
 136
 137$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 138
 139## Scalar resolute of $\boldsymbol{a}$ on $\boldsymbol{b}$
 140
 141$$r_s = |\boldsymbol{u}| = \boldsymbol{a} \cdot \hat{\boldsymbol{b}}$$
 142
 143## Vector resolute of $\boldsymbol{a} \perp \boldsymbol{b}$
 144
 145$$\boldsymbol{w} = \boldsymbol{a} - \boldsymbol{u} \> \text{ where } \boldsymbol{u} \text{ is projection } \boldsymbol{a} \text{ on } \boldsymbol{b}$$
 146
 147## Vector proofs
 148
 149### Concurrent lines
 150
 151$\ge$ 3 lines intersect at a single point  
 152
 153### Collinear points
 154
 155$\ge$ 3 points lie on the same line  
 156$\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 < \mu < 1$  
 157Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$
 158
 159### Useful vector properties
 160
 161- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
 162- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
 163- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
 164- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
 165
 166## Linear dependence
 167
 168Vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ are linearly dependent if they are non-parallel and:
 169
 170$$k\boldsymbol{a}+l\boldsymbol{b}+m\boldsymbol{c} = 0$$
 171$$\therefore \boldsymbol{c} = m\boldsymbol{a} + n\boldsymbol{b} \quad \text{(simultaneous)}$$
 172
 173$\boldsymbol{a}, \boldsymbol{b},$ and $\boldsymbol{c}$ 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.
 174
 175Vector $\boldsymbol{w}$ is a linear combination of vectors $\boldsymbol{v_1}, \boldsymbol{v_2}, \boldsymbol{v_3}$
 176
 177## Three-dimensional vectors
 178
 179Right-hand rule for axes: $z$ is up or out of page.
 180
 181i![](graphics/vectors-3d.png)
 182
 183## Parametric vectors
 184
 185Parametric equation of line through point $(x_0, y_0, z_0)$ and parallel to $a\boldsymbol{i} + b\boldsymbol{j} + c\boldsymbol{k}$ is:
 186
 187\begin{equation}\begin{cases}x = x_o + a \cdot t \\ y = y_0 + b \cdot t \\ z = z_0 + c \cdot t\end{cases}\end{equation}
 188
 189## Vector functions
 190
 191$$\boldsymbol{r}(t)=x\boldsymbol{i}+y\boldsymbol{y}$$
 192
 193- If $\boldsymbol{r}(t)$ represents position with time, then the graph of endpoints of $\boldsymbol{r}(t)$ represents the Cartesian path.
 194- Domain of $\boldsymbol{r}(t)$ is the range of $x(t)$
 195- Range of $\boldsymbol{r}(t)$ is the range of $y(t)$
 196
 197## Vector calculus
 198
 199### Derivative
 200
 201Let $\boldsymbol{r}(t)=x(t)\boldsymbol{i} + y(t)\boldsymmbol(j)$. If both $x(t)$ and $y(t)$ are differentiable, then:
 202
 203$$\boldsymbol{r}(t)=x(t)\boldsymbol{i}+y(t)\boldsymbol{j}$$