物体の運動の表し方ベクトルとその内積単位ベクトル、基本ベクトル、大きさ、直交

学習環境

物理学講義力学 (松下貢(著)、裳華房)の第1章(物体の運動の表し方)、1.4(ベクトルとその内積)の問題6の解答を求めてみる。

\begin{array}{l} {| i |}^{2} \\ = i \cdot i \\ = (1, 0, 0) \cdot (1, 0, 0) \\ = 1 \end{array}

\begin{array}{l} {| j |}^{2} \\ = j \cdot j \\ = (0, 1, 0) \cdot (0, 1, 0) \\ = 1 \end{array}

\begin{array}{l} {| k |}^{2} \\ = k \cdot k \\ = (0, 0, 1) \cdot (0, 0, 1) \\ = 1 \end{array}

よって、

{| i |}^{2} = {| j |}^{2} = {| k |}^{2} = 1

また、

\begin{array}{l} i \cdot j = (1, 0, 0) \cdot (0, 1, 0) = 0 \\ j \cdot k = (0, 1, 0) \cdot (0, 0, 1) = 0 \\ k \cdot i = (0, 0, 1) \cdot (1, 0, 0) = 0 \end{array}

よって、

i \cdot j = j - k = k - i = 0

コード

#!/usr/bin/env python3
from unittest import TestCase, main
from sympy import Matrix

print('6.')

i = Matrix([1, 0, 0])
j = Matrix([0, 1, 0])
k = Matrix([0, 0, 1])
vs = [i, j, k]


class Test(TestCase):
    def test_norm(self):
        for v in vs:
            self.assertEqual(v.norm() ** 2, 1)

    def test_orthogonal(self):
        for v in vs:
            for w in vs:
                if v == w:
                    continue
                self.assertEqual(v.dot(w), 0)


if __name__ == "__main__":
    main()

入出力結果

% ./sample6.py -v
6.
test_norm (__main__.Test) ... ok
test_orthogonal (__main__.Test) ... ok

----------------------------------------------------------------------
Ran 2 tests in 0.002s

OK
%