{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "[2015M217](http://colliand.com/2015M217/)\n", "\n", "#Vectors in Space\n", "\n", "* J. Colliander\n", "* Stewart Text: 12.[1-5]\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "##Where are you? Where are you going?" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo(\"zBlAGGzup48\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* $\\mathbb{R}^3$\n", " * Cartesian coordinate system, right hand rule, coordinate planes\n", " * Distance formula\n", " * Sphere, Cylinder\n", " * Other coordinate systems\n", "\n", "![Cartesian Coordinates](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fd/3D_Vector.svg/300px-3D_Vector.svg.png)\n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true, "slideshow": { "slide_type": "slide" } }, "source": [ "\n", "* Vectors ${\\bf{a}} = (a_x, a_y, a_z)$.\n", "\t* magnitude and direction\n", " * Vector Space\n", " * vector addition\n", " * ${\\bf{0}}$ vector\n", " * multiplication by a scalar\n", "\t* standard basis, unit vectors $\\bf{i}, \\bf{j}, \\bf{k}$, \n", " * $\\mathbb{R}^d$ has standard basis $\\{ e^j \\}_{j=1}^d$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Dot Product\n", "\n", "The `Dot Product` takes two vectors and returns a scalar: ${\\bf{a}}, {\\bf{b}} \\longmapsto {\\bf{a}}\\cdot {\\bf{b}}$.\n", "\n", "$${\\bf{a}}=\\langle a_1, a_2, a_3 \\rangle, {\\bf{b}}=\\langle b_1, b_2, b_3 \\rangle$$\n", "\n", "$${\\bf{a}} \\cdot {\\bf{b}} = a_1 b_1 + a_2 b_2 + a_3 b_3 $$\n", "\n", "`Direction Cosines`\n", "\n", "## Properties\n", "\n", "* `Orthogonal` vectors\n", "\n", "${\\bf{a}} \\perp {\\bf{b}}$, read ${\\bf{a}}$ is *orthogonal* to ${\\bf{b}}$, if ${\\bf{a}} \\cdot {\\bf{b}} = 0$.\n", "\n", "* cosine property\n", "\n", "${\\bf{a}} \\cdot {\\bf{b}} = |{\\bf{a}}||{\\bf{b}}| \\cos (\\theta).$ \n", "\n", "* *projections*\n", "\n", "![projection](https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Dot_Product.svg/300px-Dot_Product.svg.png)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#Cross Product\n", "\n", "The *Cross Product* takes two vectors and returns a vector: \n", "\n", "![cross product](https://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Cross_product_parallelogram.svg/220px-Cross_product_parallelogram.svg.png)\n", "\n", "##Properties\n", "\n", "* Formula: $ {\\bf{a}} \\times {\\bf{b}} = \\langle a_2 b_3 - a_3 b_2, a_3 b_1 - a_1 b_3, a_1 b_2 - a_2 b_1 \\rangle $\n", "* Determinants\n", "* Parallelogram\n", "\n", "*Scalar Triple Product*\n", "\n", "* Parallelepiped\n", "\n", "![parallelepiped](https://upload.wikimedia.org/wikipedia/en/thumb/e/e6/Exterior_calc_triple_product.svg/220px-Exterior_calc_triple_product.svg.png)" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "a = np.array([4,7,3]); b=np.array([0,1,1])" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "10" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.inner(a, b)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 4, -4, 4])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.cross(a,b)\n" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "c = np.array([1,1,1])" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "4" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.inner(np.cross(a,b),c)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Equations of Lines and Planes" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "In $\\mathbb{R}^2$, we have the equation for a `line`:\n", "$$ ax + by = c.$$\n", "Of course, we can solve for $y$ to rewrite the equation in `slope-intercept form`:\n", "$$by = c - ax$$\n", "$$y = -\\frac{a}{b} x + \\frac{c}{b}.$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In $\\mathbb{R}^3$, we can form a similar expression and this describes a `plane`:\n", "$$ a x + by + cz = d.$$\n", "We can solve for $z$:\n", "$$cz = d -ax - by$$\n", "$$z = -\\frac{a}{c} x - \\frac{b}{c}y + \\frac{d}{c}$$\n", "\n", "Another natural way to express a `plane` is to normalize so that $d=1$. You can do this by dividing through by $d$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Natural Questions!**\n", "\n", "* Which planes pass through the coordinate origin?\n", "* Can you describe the points on a plane with equation $ax + by + cz = d$ which are also in the $(x,y)$-plane described by $z = 0$?\n", "* Find the equation for the plane passing through three points.\n", "* Find the equation passing through the point $P$ with coordinates $(f,g,h)$ which is perpindicular to ${\\bf{n}} = (1,2,3)$." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.4.3" } }, "nbformat": 4, "nbformat_minor": 0 }