学習環境
- Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
- Windows 10 Pro (OS)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
ラング線形代数学(上)(S.ラング (著)、芹沢 正三 (翻訳)、ちくま学芸文庫)の1章(R^n におけるベクトル)、3(ベクトルのノルム)、練習問題10、11.を取り組んでみる。
cosθ = 1 (0 ≤ θ ≤ 1)のならば、θ = 0となるので、ベクトルAとBは同じ向きを持つ。
cosθ = 0 (0 ≤ θ ≤ 1)のならば、θ = πとなるので、ベクトルAとBは反対の向きを持つ。
距離の可換性について。
三角不等式について。
コード(Emacs)
Python 3
#!/usr/bin/env python3 # -*- coding: utf-8 -*- from sympy import pprint, symbols, Matrix, solve print('11.') A = Matrix(symbols('a1 a2', real=True)) B = Matrix(symbols('b1 b2', real=True)) C = Matrix(symbols('c1 c2', real=True)) print((A - B).norm() == (B - A).norm()) X = (A - B).norm() Y = (A - C).norm() + (B - C).norm() for t in [X, Y]: pprint(t) print() pprint(X <= Y)
入出力結果(Terminal, Jupyter(IPython))
$ ./sample10.py 11. True _________________________ ╱ 2 2 ╲╱ (a₁ - b₁) + (a₂ - b₂) _________________________ _________________________ ╱ 2 2 ╱ 2 2 ╲╱ (a₁ - c₁) + (a₂ - c₂) + ╲╱ (b₁ - c₁) + (b₂ - c₂) _________________________ _________________________ _____________ ╱ 2 2 ╱ 2 2 ╱ 2 ╲╱ (a₁ - b₁) + (a₂ - b₂) ≤ ╲╱ (a₁ - c₁) + (a₂ - c₂) + ╲╱ (b₁ - c₁) + ____________ 2 (b₂ - c₂) $
HTML5
<div id="graph0"></div> <pre id="output0"></pre> <label for="r0">r = </label> <input id="r0" type="number" min="0" value="0.5"> <label for="dx">dx = </label> <input id="dx" type="number" min="0" step="0.0001" value="0.001"> <br> <label for="x1">x1 = </label> <input id="x1" type="number" value="-5"> <label for="x2">x2 = </label> <input id="x2" type="number" value="5"> <br> <label for="y1">y1 = </label> <input id="y1" type="number" value="-5"> <label for="y2">y2 = </label> <input id="y2" type="number" value="5"> <br> <label for="a1">a1 = </label> <input id="a1" type="number" value="2"> <label for="a2">a2 = </label> <input id="a2" type="number" value="3"> <br> <label for="b1">b1 = </label> <input id="b1" type="number" value="-4"> <label for="b2">b2 = </label> <input id="b2" type="number" value="5"> <button id="draw0">draw</button> <button id="clear0">clear</button> <script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script> <script src="sample10.js"></script>
JavaScript
let div0 = document.querySelector('#graph0'), pre0 = document.querySelector('#output0'), width = 600, height = 600, padding = 50, btn0 = document.querySelector('#draw0'), btn1 = document.querySelector('#clear0'), input_r = document.querySelector('#r0'), input_dx = document.querySelector('#dx'), input_x1 = document.querySelector('#x1'), input_x2 = document.querySelector('#x2'), input_y1 = document.querySelector('#y1'), input_y2 = document.querySelector('#y2'), input_a1 = document.querySelector('#a1'), input_a2 = document.querySelector('#a2'), input_b1 = document.querySelector('#b1'), input_b2 = document.querySelector('#b2'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_a1, input_a2, input_b1, input_b2], p = (x) => pre0.textContent += x + '\n', range = (start, end, step=1) => { let res = []; for (let i = start; i < end; i += step) { res.push(i); } return res; }; let draw = () => { pre0.textContent = ''; let r = parseFloat(input_r.value), dx = parseFloat(input_dx.value), x1 = parseFloat(input_x1.value), x2 = parseFloat(input_x2.value), y1 = parseFloat(input_y1.value), y2 = parseFloat(input_y2.value), a1 = parseFloat(input_a1.value), a2 = parseFloat(input_a2.value), b1 = parseFloat(input_b1.value), b2 = parseFloat(input_b2.value); if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) { return; } let points = [], lines = [[0, 0, a1, a2, 'red'], [0, 0, b1, b2, 'green'], [a1, a2, b1, b2, 'blue']], fns = [], fns1 = [], fns2 = []; fns .forEach((o) => { let [f, color] = o; for (let x = x1; x <= x2; x += dx) { let y = f(x); points.push([x, y, color]); } }); fns2 .forEach((o) => { let [f, color] = o; for (let x = x1; x <= x2; x += dx0) { let g = f(x); lines.push([x1, g(x1), x2, g(x2), color]); } }); let xscale = d3.scaleLinear() .domain([x1, x2]) .range([padding, width - padding]); let yscale = d3.scaleLinear() .domain([y1, y2]) .range([height - padding, padding]); let xaxis = d3.axisBottom().scale(xscale); let yaxis = d3.axisLeft().scale(yscale); div0.innerHTML = ''; let svg = d3.select('#graph0') .append('svg') .attr('width', width) .attr('height', height); svg.selectAll('line') .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines)) .enter() .append('line') .attr('x1', (d) => xscale(d[0])) .attr('y1', (d) => yscale(d[1])) .attr('x2', (d) => xscale(d[2])) .attr('y2', (d) => yscale(d[3])) .attr('stroke', (d) => d[4] || 'black'); svg.selectAll('circle') .data(points) .enter() .append('circle') .attr('cx', (d) => xscale(d[0])) .attr('cy', (d) => yscale(d[1])) .attr('r', r) .attr('fill', (d) => d[2] || 'green'); svg.append('g') .attr('transform', `translate(0, ${height - padding})`) .call(xaxis); svg.append('g') .attr('transform', `translate(${padding}, 0)`) .call(yaxis); [fns, fns1, fns2].forEach((fs) => p(fs.join('\n'))); }; inputs.forEach((input) => input.onchange = draw); btn0.onclick = draw; btn1.onclick = () => pre0.textContent = ''; draw();
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