spec / vectors.mdon commit geometric proofs with vectors (2183c3a)
   1---
   2header-includes:
   3  - \documentclass{standalone}
   4  - \usepackage{cleveref}
   5  - \usepackage{harpoon}
   6  - \usepackage{accent}
   7  - \usepackage{amsmath}
   8...
   9
  10# Vectors
  11
  12- **vector:** a directed line segment  
  13- arrow indicates direction
  14- length indicates magnitude
  15- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
  16- column notation: $\begin{bmatrix}
  17       x \\ y
  18     \end{bmatrix}$
  19- vectors with equal magnitude and direction are equivalent
  20
  21
  22![](graphics/vectors-intro.png)
  23
  24## Vector addition
  25
  26$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.  
  27Addition is commutative (parallelogram)
  28
  29## Scalar multiplication
  30
  31For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$.
  32
  33When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.
  34
  35## Vector subtraction
  36
  37To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$
  38
  39## Parallel vectors
  40
  41Parallel vectors have same direction or opposite direction.
  42
  43**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**
  44
  45## Position vectors
  46
  47Vectors may describe a position relative to $O$.
  48
  49For a point $A$, the position vector is $\boldsymbol{OA}$
  50
  51## Linear combinations of non-parallel vectors
  52
  53If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:
  54
  55$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$
  56
  57## Column vector notation
  58
  59A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}$
  60
  61## Component notation
  62
  63A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.  
  64$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$  
  65Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$
  66
  67Basic algebra applies:  
  68$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  
  69Two vectors equal if and only if their components are equal.
  70
  71## Unit vectors
  72
  73A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.
  74
  75A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:
  76
  77$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$
  78
  79Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$
  80
  81## Scalar products / dot products
  82
  83If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is:
  84$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$
  85
  86Produces a real number, not a vector.
  87
  88$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$
  89
  90## Geometric scalar products
  91
  92$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$
  93
  94where $0 \le \theta \le \pi$
  95
  96## Perpendicular vectors
  97
  98If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)
  99
 100## Finding angle between vectors
 101
 102$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$
 103
 104
 105## Vector projections
 106
 107Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.
 108
 109$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$
 110
 111## Vector proofs
 112
 113**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
 114**Collinear points -** $\ge$ 3 points lie on the same line
 115
 116Useful vector properties:
 117
 118- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
 119- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
 120- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
 121- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$
 122
 123
 124
 125