From 3387605b4537c8b42a50613df825c6daf6a35ca4 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Thu, 18 Oct 2018 10:35:30 +1100 Subject: [PATCH] vector projections & signed lengths, changed \vec to \boldsymbol --- spec/vectors.md | 53 +++++++++++++++++++++++++++++-------------------- 1 file changed, 31 insertions(+), 22 deletions(-) diff --git a/spec/vectors.md b/spec/vectors.md index f832af7..a2122b0 100644 --- a/spec/vectors.md +++ b/spec/vectors.md @@ -3,8 +3,9 @@ header-includes: - \documentclass{standalone} - \usepackage{cleveref} - \usepackage{harpoon} - - \usepackage{accent} \newcommand{\vect}[1]{\accentset{\rightharpoonup}{#1}} ---- + - \usepackage{accent} + - \usepackage{amsmath} +... # Vectors @@ -22,36 +23,36 @@ header-includes: ## Vector addition -$\vec{u} + \vec{v}$ can be represented by drawing each vector head to tail then joining the lines. +$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines. Addition is commutative (parallelogram) ## Scalar multiplication -For $k \in \mathbb{R}^+$, $k\vec{u}$ has the same direction as $\vec{u}$ but length is multiplied by a factor of $k$. +For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$. When multiplied by $k < 0$, direction is reversed and length is multplied by $k$. ## Vector subtraction -To find $\vec{u} - \vec{v}$, add $\vec{-v}$ to $\vec{u}$ +To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$ ## Parallel vectors Parallel vectors have same direction or opposite direction. -**Two non-zero vectors $\vec{u}$ and $\vec{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\vec{u} = k \vec{v}$** +**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$** ## Position vectors Vectors may describe a position relative to $O$. -For a point $A$, the position vector is $\vec{OA}$ +For a point $A$, the position vector is $\boldsymbol{OA}$ ## Linear combinations of non-parallel vectors -If two non-zero vectors $\vec{a}$ and $\vec{b}$ are not parallel, then: +If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then: -$$m\vec{a} + n\vec{b} = p \vec{a} + q \vec{b}\quad\text{implies}\quad m = p, \> n = q$$ +$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad\text{implies}\quad m = p, \> n = q$$ ## Column vector notation @@ -59,45 +60,53 @@ A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begi ## Component notation -A vector $\vec{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\vec{u} = x\vec{i} + y\vec{j}$. -$\vec{u}$ is the sum of two components $x\vec{i}$ and $y\vec{j}$ -Magnitude of vector $\vec{u} = x\vec{i} + y\vec{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$ +A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$. +$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$ +Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$ Basic algebra applies: -$(x\vec{i} + y\vec{j}) + (m\vec{i} + n\vec{j}) = (x + m)\vec{i} + (y+n)\vec{j}$ +$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$ Two vectors equal if and only if their components are equal. ## Unit vectors -A vector of length 1. $\vec{i}$ and $\vec{j}$ are unit vectors. +A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors. + +A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$: -A unit vector in direction of $\vec{a}$ is denoted by $\hat{\vec{a}}$ +$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$ -Also, unit vector of $\vec{a}$ can be defined by $\vec{a} \cdot {|\vec{a}|}$ +Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$ ## Scalar products / dot products -If $\vec{a} = a_i \vec{i} + a_2 \vec{j}$ and $\vec{b} = b_i \vec{i} + b_2 \vec{j}$, the dot product is: -$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2$$ +If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is: +$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$ Produces a real number, not a vector. -$$\vec{a} \cdot \vec{a} = |\vec{a}|^2$$ +$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$ ## Geometric scalar products -$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$ +$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$ where $0 \le \theta \le \pi$ ## Perpendicular vectors -If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$ (since $\cos 90 = 0$) +If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$) ## Finding angle between vectors -$$\cos \theta = {{\vec{a} \cdot \vec{b}} \over {|\vec{a}| |\vec{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\vec{a}| |\vec{b}|}}$$ +$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$ ## Vector projections +Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$. + +$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$ + + + -- 2.43.2