a∥b ⟺ a=λb ⟺ x1y2=x2y1\boldsymbol{a} \parallel \boldsymbol{b} \iff \boldsymbol{a} = \lambda \boldsymbol{b} \iff x_1y_2 = x_2y_1a∥b⟺a=λb⟺x1y2=x2y1
cos⟨a,b⟩=a⋅b∣a∣∣b∣=x1x2+y1y2x12+y12x22+y22\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}}cos⟨a,b⟩=∣a∣∣b∣a⋅b=x12+y12x22+y22x1x2+y1y2
a⊥b ⟺ a⋅b=0 ⟺ x1x2+y1y2=0\boldsymbol{a} \perp \boldsymbol{b} \iff \boldsymbol{a} \cdot \boldsymbol{b} = 0 \iff x_1x_2 + y_1y_2 = 0a⊥b⟺a⋅b=0⟺x1x2+y1y2=0