Hi, I am having trouble with this proof, I am wondering what to do because I keep getting the wrong answer.
I want to prove
||x-y|| = ||x|| ||y|| ||x bar - y bar||
where
x and y are vectors in Rn
and u bar is defined by u/(||u||^2)
The way i started it was expanding the right side completely by using the norm formula and then simplify it into making it equal to the left, but I end up with the wrong answer and a really messy calculation. I was wondering if there was a simpler way to do this.
Well this is what the question asks exactly
Given a nonzero vector u E R^n, put u bar as:
(1/||u||^2)*u
and the rest is the same, equality you put is correct, that is what it asks me to prove.
I figured that (1/||u||^2)*u would be the same as u/||u||^2, the way the question writes it is (1/||u||^2)*u.
Any ideas as to what is missing? When I try to do the proof by going the long hard way and expanding the norm of x bar - y bar, I get close to the left side but its not quite there after simplifying everything.
Ok so this is what i did for the norm of x bar - y bar
if x = (x1,...,xn) and y = (y1,...,yn)
then i have
||(x1,...,xn)/[sqrt(x1^2+...+xn^2)]^2 - (y1,...,yn)/[sqrt(y1^2+...+yn^2)]^2||
im just going to denote sqrt(x1^2+...+xn^2) = c and sqrt(y1^2+...+yn^2) = d
=sqrt[ (x1/(c^2) - y1/(d^2))^2 +...+(xn/(c^2) - yn/(d^2))^2]
and I went from there onward by simplifying. Is that the right way to do it?