high School⏱ 3 min

Dot Product

Geometric meaning of dot product

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u=(2,1)v=(1,2)36.9°u·v = 4

u = (2,1)

v = (1,2)

u⃗⋅v⃗=uxvx+uyvy=2⋅1+1⋅2=4\vec{u}\cdot\vec{v} = u_x v_x + u_y v_y = 2\cdot1+1\cdot2 = 4
cos⁡θ=42.24⋅2.24⇒θ=36.9°\cos\theta = \frac{4}{2.24\cdot2.24} \Rightarrow \theta = 36.9°

Dot product = measures how "aligned" two vectors are. u·v=0 → perpendicular, u·v>0 → acute angle, u·v<0 → obtuse angle. Foundation of projections.