# Thread: Using the dot product in a proof

1. ## Using the dot product in a proof

So I have to prove that for any 2 vectors x & y:

|x - y| >= ||x| - |y||

and

|x - y|^2 + |x + y|^2 = 2(|x|^2 + |y|^2)

I am familiar with an algebraic proof for the first part however they need to be proved specifically using the dot product rule. Also in geometry what implications might the second equation have with regards to parallelograms?

2. Originally Posted by kevinlightman
So I have to prove that for any 2 vectors x & y:

|x - y| >= ||x| - |y||

and

|x - y|^2 + |x + y|^2 = 2(|x|^2 + |y|^2)

I am familiar with an algebraic proof for the first part however they need to be proved specifically using the dot product rule. Also in geometry what implications might the second equation have with regards to parallelograms?
$\displaystyle |x-y|^2=(x-y).(x-y)=|x|^2-x.y-y.x+|y|^2\ \ \ \ \ ...(1)$

ans:

$\displaystyle ||x|-|y||=|x|^2-2|x||y|+|y|^2\ \ \ \ \ ...(2)$

but $\displaystyle x.y\le|x||y|$ hence $\displaystyle (1)\ \ge\ (2)$

CB