數學之旅：三角形面積公式(Ⅳ)
數學之旅：三角形面積公式(Ⅳ) (Mathematical Journey through the Formulas of Triangle Area)
國立蘭陽女中 陳敏晧教師
當數學旅程來到空間時，我們首先需要空間向量的外積(cross product)：兩空間向量 \(\vec{a} = \left( {{x_1},{y_1},{z_1}} \right),\vec{b} = \left( {{x_2},{y_2},{z_2}} \right)\) 的外積定義為
\(\begin{array}{ll}\vec{n} &= \vec{a} \times \vec{b} = \left( {\left| {\begin{array}{*{20}{c}} {{y_1}}&{{z_1}}\\ {{y_2}}&{{z_2}} \end{array}} \right|,\left| {\begin{array}{*{20}{c}} {{z_1}}&{{x_1}}\\ {{z_2}}&{{x_2}} \end{array}} \right|,\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}\\ {{x_2}}&{{y_2}} \end{array}} \right|} \right) \\&= \left( {{y_1}{z_2} – {y_2}{z_1},{x_2}{z_1} – {x_1}{z_2},{x_1}{y_2} – {x_2}{y_1}} \right)\end{array}\)
外積有三個性質：