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 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