Thanks for any help u t2 11121122 llell2 LLU U Solutionu

Thanks for any help!

(u . t,)2 (1112-1-12)/2 + llell2 LL.\'U U

Solution

u + v(u.v) = w

First take dot product of u on both sides of given equation we get

u.u + (u.v)^2 = u.w

||u||^2 + (u.v)^2 = u.w -----(1)

Now take original equation :

u + v(u.v) = w

v(u.v) = w - u

Take square on both sides:

{v(u.v)}^2 = ( w - u)^2

v.v(u.v)^2 = w^2 + u^2 - 2u.w

= ||w||^2 + ||u||^2 - 2u.w

Replace u.w from equation 1:

||v||^2(u.v)^2 = ||w||^2 + ||u||^2 - 2(||u||^2 + (u.v)^2)

(u.v)^2(||v||^2 +2) = ||w||^2 - ||u||^2

Divide both sides by (||v||^2 +2):

(u.v)^2 = [ ||w||^2 - ||u||^2 ]/(||v||^2 +2)

Hence proved