{"metadata": {"name": "", "signature": "sha256:5a9317d71f0a72fc50c9be61898caa613606100e3965c2716f401b5dfcf9523d"}, "nbformat": 3, "nbformat_minor": 0, "worksheets": [{"metadata": {}, "cells": [{"metadata": {}, "source": ["# Making vectors orthogonal"], "cell_type": "markdown"}, {"input": ["import numpy as np\n", "import matplotlib.pyplot as pt"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 4, "cell_type": "code", "outputs": []}, {"input": ["np.random.seed(13)\n", "x = np.random.randn(2)\n", "y = np.random.randn(2)"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 25, "cell_type": "code", "outputs": []}, {"input": ["pt.arrow(0, 0, x[0], x[1], color=\"blue\")\n", "pt.arrow(0, 0, y[0], y[1], color=\"red\")\n", "pt.xlim([-1, 1])\n", "pt.ylim([-1, 1])\n", "pt.gca().set_aspect(\"equal\")\n", "pt.grid()"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 22, "cell_type": "code", "outputs": [{"metadata": {}, "text": ["<matplotlib.figure.Figure at 0x7fecc016feb8>"], "output_type": "display_data", "png": 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{"metadata": {}, "source": ["Are those orgonal? How would we find out?"], "cell_type": "markdown"}, {"input": ["x.dot(y)"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 26, "cell_type": "code", "outputs": [{"metadata": {}, "text": ["0.37226452785072001"], "prompt_number": 26, "output_type": "pyout"}]}, {"metadata": {}, "source": ["Now use the formula to make `ynew` which *is* orthogonal to `x`:"], "cell_type": "markdown"}, {"input": ["ynew = y - y.dot(x)/x.dot(x)*x"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 27, "cell_type": "code", "outputs": []}, {"input": ["pt.arrow(0, 0, x[0], x[1], color=\"blue\")\n", "pt.arrow(0, 0, ynew[0], ynew[1], color=\"red\")\n", "pt.xlim([-1, 1])\n", "pt.ylim([-1, 1])\n", "pt.gca().set_aspect(\"equal\")\n", "pt.grid()"], "metadata": {}, "language": "python", "collapsed": false, "prompt_number": 28, "cell_type": "code", "outputs": [{"metadata": {}, "text": ["<matplotlib.figure.Figure at 0x7fecbb501630>"], "output_type": "display_data", "png": 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{"input": [], "metadata": {}, "language": "python", "collapsed": false, "cell_type": "code", "outputs": []}]}]}