（1）平行的充要条件

$$\boldsymbol{a} \parallel \boldsymbol{b} \iff \boldsymbol{a} = \lambda \boldsymbol{b} \iff x_1y_2 = x_2y_1$$

（2）夹角的余弦公式

$$\cos\langle \boldsymbol{a},\boldsymbol{b} \rangle = \frac{\boldsymbol{a} \cdot \boldsymbol{b}}{|\boldsymbol{a}||\boldsymbol{b}|} = \frac{x_1x_2 + y_1y_2}{\sqrt{x_1^2 + y_1^2}\sqrt{x_2^2 + y_2^2}}$$

（3）垂直的充要条件

$$\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0 \iff x_1x_2 + y_1y_2 = 0$$