推导一个向量逆时针绕起点旋转α度后得到的向量

向量(x, y)逆时针绕起点旋转$\alpha$度后得到的向量(x’, y’)：

$x' = xcos\alpha - ysin\alpha$
$y' = xsin\alpha+ ycos\alpha$

推导过程：

$d = \sqrt{x^2+y^2}$

$cos\theta = x/d$
$sin\theta = y/d$

$cos(\theta+\alpha) = x' /d$
$sin(\theta+\alpha) = y' /d$

由：
$cos(\alpha+\theta) = cos\alpha cos\theta - sin\alpha sin\theta$
$sin(\alpha+\theta) = sin\alpha cos\theta + cos\alpha sin\theta$

得：
$cos(\theta+\alpha) = cos\alpha cos\theta - sin\alpha sin\theta$

$= cos\alpha\frac{x}{d} - sin\alpha \frac{y}{d} = \frac{ x'}{d}$

$sin(\theta+\alpha) = sin\alpha cos\theta + cos\alpha sin\theta$

$= sin\alpha\frac{x}{d} + cos\alpha \frac{y}{d} = \frac{ y'}{d}$

消除$d$得：
$x' = cos\alpha\cdot x - sin\alpha\cdot y$

$y' = sin\alpha\cdot x + cos\alpha\cdot y$

文章来源: https://blog.csdn.net/liuzhuomei0911/article/details/92250989