0. 记号
$\mathbf a$或$\vec a$表示向量
$\mathbf{i, j}$分别表示$x$方向、$y$方向的单位向量
1. 叉积
$\mathbf{i \times j} = 1\ \mathbf{j \times i} = 0 \ \mathbf{a \times a} = 0$
$\mathbf{a \times b = |a||b|}\sin \theta = (x_1\mathbf i + y_1\mathbf j) \times (x_1\mathbf i + y_1\mathbf j) = x_1 y_2 - x_2 y_1$
2. 点积
$\mathbf{i \cdot j} = 0\ \mathbf{j \cdot i} = 0 \ \mathbf{a \cdot a} = |\mathbf a| ^ 2$
$\mathbf{a \cdot b = |a||b|}\cos \theta = (x_1\mathbf i + y_1\mathbf j) \cdot (x_1\mathbf i + y_1\mathbf j) = x_1 x_2 + y_1 y_2$
3. 和角公式
设$\mathbf a = (\cos \alpha, \sin \alpha), \mathbf b = (\cos \beta, \sin \beta), \beta > \alpha$
$|\mathbf a| = |\mathbf b| = 1$
余弦
$\mathbf{a \cdot b = |a||b|}\cos(\beta - \alpha) = \cos(\beta - \alpha)$
$\mathbf{a \cdot b = }\cos \alpha \cos \beta + \sin \alpha \sin \beta$
$\cos (\alpha - \beta) = \cos (\beta - \alpha) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$
$\cos(\alpha + \beta) = \cos (\alpha - (-\beta)) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$
正弦
$\mathbf{a \times b = |a||b|}\sin (\beta - \alpha) = \sin(\beta - \alpha)$
$\mathbf{a \times b} = \cos \alpha\sin \beta - \cos \beta \sin \alpha$
$\sin(\alpha - \beta) = - \sin(\beta - \alpha) = \cos \beta \sin \alpha - \cos \alpha \sin \beta$
$\sin(\alpha + \beta) = \sin(\alpha - (-\beta)) = \cos \beta \sin \alpha + \cos \alpha \sin \beta$
4. 向量旋转
设$\mathbf a = (x, y) = k(\cos \alpha, \sin \alpha)$逆时针旋转$\theta$至$\mathbf b$
$\mathbf b = k(\cos(\alpha + \theta), \sin(\alpha + \theta)) = k(\cos \alpha \cos \theta - \sin \alpha \sin \theta, \sin \alpha \cos \theta + \sin \theta \cos \alpha) = (x \cos \theta - y \sin \theta, y \cos \theta + x \sin \theta)$
5. 切比雪夫距离与曼哈顿距离
与某个点切比雪夫距离为1的点围成了一个正着的正方形，曼哈顿距离为一个斜着的正方形。切比雪夫边长是曼哈顿的$\sqrt 2$倍，并且旋转了$\frac \pi 4$。
切->曼：$(x, y) \to \frac{\sqrt 2}{2}(x \cos \frac \pi 4 - y \sin \frac \pi 4, x \cos \frac \pi 4 + y \sin \frac \pi 4) = (\frac{x - y} 2, \frac{x + y} 2)$
曼->切：$(x, y) \to \sqrt 2(x \cos \frac \pi 4 - y \sin \frac \pi 4, x \cos \frac \pi 4 + y \sin \frac \pi 4) = (x - y, x + y)$