Suppose that u is a non-zero vector in R^n. Prove that ||u-v|| = ||u|| - ||v|| if and only if v=ku for k element of R with 0 <= k <= 1.
I tried putting ku in the equation but seem to have hit a road block
any ideas?
Well if $\displaystyle v=ku$ then
$\displaystyle ||u-v||=||u-ku||=||u(1-k)||=(1-k)||u||=||u||-k||u||=||u||-||ku||=||u||-||v||$.
For the converse, it depends; is $\displaystyle ||\bullet||$ the standard norm in $\displaystyle \mathbb{R}^n$ or a norm with respect to an arbitrary inner product? What are you familiar with? You have to be specific.