SOLUTION: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||.

Algebra ->  Vectors -> SOLUTION: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||.      Log On


   

Question 1203666: Show that 4u · v = ||u + v||^2 − ||u − v||^2. Show that u and v is orthogonal if and only if ||u + v|| = ||u − v||.
Answer by ikleyn(52779) About Me  (Show Source):
You can put this solution on YOUR website!
.
(a) Show that 4u · v = ||u + v||^2 − ||u − v||^2.
(b) Show that u and v highlight%28cross%28is%29%29 highlight%28highlight%28are%29%29 orthogonal if and only if ||u + v|| = ||u − v||.
~~~~~~~~~~~~~~~~~~ 

             Part (a)


    ||u + v||^2 = (u,u) + 2(u,v) + (v,v)      (1)

    ||u - v||^2 = (u,u) - 2(u,v) + (v,v)      (2)


Here (u,v) is the scalar product of vectors u and v, or, in your designations, (u,v) = u · v.


Subtract equation (2) from equation (1) .  You will get 

    ||u + v||^2 − ||u − v||^2 = 4u · v.


It is exactly what should be proven in part (a).



             Part (b)


Vectors u and v are orthogonal if and only if  u · v = 0.


In part (a), we show that  4u · v = ||u + v||^2 − ||u − v||^2.


So, u · v = 0  if and only if  ||u + v||^2 − ||u − v||^2 = 0,  or  ||u + v||^2 = ||u − v||^2.


It is exactly what should be proven in part (b).

Solved, proved and completed.