向量的叉积与点积

最后更新于:2022-04-02 04:09:44

[TOC] ## 向量叉积 ### 2d-向量叉积 `$ (x_{1},y_{1})\times(x_{2},y_{2})= \begin{vmatrix} x_{1} & x_{2} \\ y_{1} & y_{2} \\ \end{vmatrix} =x_{1}y_{2}-x_{2}y_{1} $` 中间的`$ \begin{vmatrix} x_{1} & x_{2} \\ y_{1} & y_{2} \\ \end{vmatrix} $`称为行列式 如: `(1,0)x(0,1)=1-0=1` `(0,1)x(1,0)=0-1=-1` 叉积在坐标系中就是两个向量的面积 #### 叉积的几何定义 `$ \vec{a} \times \vec{b} = \left | \vec{a} \right | \times \left | \vec{b} \right | \times sine \theta $` ### 3d -向量叉积 ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/0d/e3/0de32461040be688c2969e93a87cce50_800x364.png) ### 行列式求值 推荐网站 https://matrix.reshish.com/determinant.php ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/e4/35/e4351e6147fa455fc3f721d8d7aefa60_184x96.png) ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/ce/55/ce555d0d2b5bbc9d419e3a7ae8828bdd_406x104.png) 行列式 `|A| = a(ei - fh) - b(di - fg) + c(dh - eg)` - 把 a 乘以不在 a 的行或列上的 2×2 矩阵的行列式 - 以 b 和 c 也做相同的计算 - 把结果加在一起,不过 b 前面有个负号 ## 向量点积 公式: `$ (x_{1},y_{1}) \cdot (x_{x},y_{2}) =x_{1}y_{2}+x_{2}y_{2} $` 如: ``` (1,0)*(0,1)=0 (2,0)*(2,0)=4 (1,2)*(2,1)=4 ``` ![](https://docs.gechiui.com/gc-content/uploads/sites/kancloud/73/1d/731d4354dcde93808f43d56f7196284a_400x383.png) 规律: `$ a \cdot b = \left | a \right | \left | b \right | cos \theta $` a,b是两个向量,`$ \theta $`是a、b的夹角;也就是a在b上的投影长度和b相乘
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