数学 - 線型代数 - 2次元と3次元の簡単な幾何学 - 直線の方程式(2)(直線、垂線の足の座標、長さ)

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線型代数入門 (松坂和夫(著)、岩波書店)の第1章(2次元と3次元の簡単な幾何学)、7(直線の方程式(2))、問7.を解いてみる。

問7.

$\begin{array}{l} a = (3, - 2) \\ q = k a = (3 k, - 2 k) \\ 3 \cdot 3 k - 2 \cdot - 2 k = 13 \\ 9 k + 4 k = 13 \\ k = 1 \\ q = (3, - 2) \\ d = \sqrt{9 + 4} = \sqrt{13} \\ q = (2, - 10) + k a = (3 k + 2, - 2 k - 10) \\ 3 (3 k + 2) - 2 (- 2 k - 10) = 13 \\ 13 k = - 13 \\ k = - 1 \\ q = (- 1, - 8) \\ d = | p - q | = | (2, - 10) - (- 1, - 8) | = | (3, - 2) | = \sqrt{13} \end{array}$
$\begin{array}{l} a = (4, 3) \\ q = k (4, 3) = (4 k, 3 k) \\ 16 k + 9 k + 18 = 0 \\ k = - \frac{25}{18} \\ q = (- \frac{50}{9}, - \frac{25}{6}) \\ d = \frac{| 18 |}{| (4, 3) |} = \frac{18}{5} \\ q = (2, - 10) + k (4, 3) = (4 k + 2, 3 k - 10) \\ 16 k + 8 + 6 k - 30 + 18 = 0 \\ 22 k = 4 \\ k = \frac{2}{11} \\ q = (\frac{30}{11}, \frac{- 104}{11}) \\ d = \frac{| (4, 3) \cdot (2, - 10) + 18 |}{| (4, 3) |} = \frac{4}{5} \end{array}$

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