Proof of Dot Product Theorem
To prove the equivalence of the two formulas for dot product we require the Law of
Cosines (a generalization of Pythagoras’ theorem):
Law of Cosines: In the triangle
c
a

b
c 2  a 2  b 2  2ab cos


Theorem: If u and v are vectors in either ℝ𝟐 or ℝ𝟑 , then
   
u  v  u v cos


where  is the smaller of the two angles between the vectors u and v when placed so that
their tails coincide.
Proof:
⃗ −v
u
⃗

u


v
We will find two different expressions for ‖u
⃗ −v
⃗ ‖2 and equate them.
By Law of Cosines, we have
‖u
⃗ −v
⃗ ‖𝟐 = ‖u
⃗ ‖2 + ‖v
⃗ ‖2 − 2‖u
⃗ ‖‖v
⃗ ‖ cos(𝜃)
By properties of the dot product, we have
‖u
⃗ −v
⃗ ‖𝟐 = (u
⃗ − v ) ⋅ (u
⃗ − v) = u
⃗ ⋅ (u
⃗ − v) − v ⋅ (u
⃗ − v)
=u
⃗ ⋅u
⃗ −u
⃗ ⋅v
⃗ −v
⃗ ⋅u
⃗ +v
⃗ ⋅v
⃗
=u
⃗ ⋅u
⃗ −u
⃗ ⋅v
⃗ −u
⃗ ⋅v
⃗ +v
⃗ ⋅v
⃗ =u
⃗ ⋅u
⃗ − 2u
⃗ ⋅v
⃗ +v
⃗ ⋅v
⃗
⃗ ‖2 − 2(u
⃗ ‖2
= ‖u
⃗ ⋅v
⃗ ) + ‖v
Equating the two expressions for ‖u
⃗ −v
⃗ ‖𝟐 we have
‖u
⃗ ‖2 + ‖v
⃗ ‖2 − 2‖u
⃗ ‖‖v
⃗ ‖ cos(𝜃) = ‖u
⃗ ‖2 − 2(u
⃗ ⋅v
⃗ ) + ‖v
⃗ ‖2
−2‖u
⃗ ‖‖v
⃗ ‖ cos(𝜃) = −2(u
⃗ ⋅v
⃗)
‖u
⃗ ‖‖v
⃗ ‖ cos(𝜃) = u
⃗ ⋅v
⃗