Why are the two definitions equivalent?

By the Law of Cosines, for u ≠ 0, v ≠ 0,

|| u - v ||2 = || u ||2 +|| v ||2 - 2 || u || || v ||cosθ

So, for u = (u1,u2,u3) and v = (v1,v2,v3),

(u1-v1)2+(u2-v2)2 + (u3-v3)2 = u12+u22+u32+v12+v22+v32 -2|| u || || v ||cosθ
Squaring the expessions on the left and simplifying,
-2u1v1-2u2v2-2u3v3
=
-2|| u || || v ||cosθ
u1v1 + u2v2+u3v3
=
|| u || || v ||cosθ
u · v
=
|| u || || v ||cosθ

If u = 0 or v = 0, both definitions immediately give u · v = 0.

Thus, the two definitions of u · v are equivalent.

dimostrazione alternativa