数学 - 線型代数 - 2次元と3次元の簡単な幾何学 - ベクトルの加法と実数倍(成分表示)

学習環境

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)、4(ベクトルの加法と実数倍)、問1-a, b.を解いてみる。

問1-a, b.

\begin{array}{l} A = (a_{1}, a_{2}), B = (b_{1}, b_{2}), C = (c_{1}, c_{2}) \\ L = (\frac{b_{1} + c_{1}}{2}, \frac{b_{2} + c_{2}}{2}) \\ M = (\frac{c_{1} + a_{1}}{2}, \frac{c_{2} + a_{2}}{2}) \\ N = (\frac{a_{1} + b_{1}}{2}, \frac{a_{2} + b_{2}}{2}) \end{array}

$\begin{array}{l} \vec{B N} + \vec{C M} \\ = (\frac{a_{1} - b_{1}}{2}, \frac{a_{2} - b_{2}}{2}) + (\frac{a_{1} - c_{1}}{2}, \frac{a_{2} - c_{2}}{2}) \\ = (a_{1} - \frac{b_{1} + c_{1}}{2}, a_{2} - \frac{b_{2} + c_{2}}{2}) \\ = \vec{L A} \end{array}$
$\begin{array}{l} \vec{B L} + \vec{C M} + \vec{A N} \\ = (\frac{c_{1} - b_{1}}{2}, \frac{c_{2} - b_{2}}{2}) + (\frac{a_{1} - c_{1}}{2}, \frac{a_{2} - c_{2}}{2}) + (\frac{b_{1} - a_{1}}{2}, \frac{b_{2} - a_{2}}{2}) \\ = (0, 0) \\ = 0 \end{array}$

Kamimura's blog