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,
If u = 0 or v = 0, both definitions immediately give u
· v = 0.
Thus, the two definitions of u · v are equivalent.
dimostrazione alternativa