a⋅b=∣a∣∣b∣cos⟨a,b⟩\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}||\boldsymbol{b}|\cos\langle \boldsymbol{a},\boldsymbol{b} \ranglea⋅b=∣a∣∣b∣cos⟨a,b⟩
a⋅b=x1x2+y1y2\boldsymbol{a} \cdot \boldsymbol{b} = x_1x_2 + y_1y_2a⋅b=x1x2+y1y2
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