Andrew's git - notes.git/blob

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