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