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  • 向量叉积

    向量叉积

    定义

    [vec a imesvec b=|vec a||vec b|sin heta ]

    证明

    enter image description here

    • 证明:在如图所示的平行四边形0ACB中 $$S_{Delta AOC}=frac{1}{2}|vec {a}||vec b|sin heta$$
    • 则平行四边形的面积是 $$S=|vec{a}| |vec b|sin heta$$

    [vec a cdot vec b=|vec a| |vec b| cos heta ]

    [cos heta=frac{vec a cdot vec b}{|vec a| |vec b|} ]

    [egin{eqnarray} sin heta &= & sqrt{1-cos^2 heta} \ &=&frac{sqrt{(|vec a|^2cdot|vec b|)^2-(vec{a}cdot{vec{b})^2}}}{|vec a||vec b|} \ end{eqnarray} ]

    [egin{aligned} S &= sqrt{(|vec a|^2cdot|vec b|)^2-(vec{a}cdot{vec{b})^2}} \ &=sqrt{(x_1^2+y_1^2)(x_2^2+y_2^2)-(x_1x_2+y_1y_2)^2} \ &= sqrt{(x_1y_2)^2+(x_2y_1)^2-2x_1x_2y_1y_2} \ &= sqrt{(x_1y_2-x_2y_1)^2} \ &= |x_1y_2-x_2y_1| \ end{aligned} ]

    $overrightarrow {} ( )underleftarrow{ssxxxxxxxxx}( )leftleftarrows{}$

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  • 原文地址:https://www.cnblogs.com/Vimin/p/11627515.html
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