# Help

• Jun 11th 2006, 04:53 PM
Susie38
Help
I don't know if this the right thread to get help, but here it goes.

Prove that:

|x + y|^2 + |x-y|^2 = 2|x|^2 + 2|y|^2 if x is an element of R^k and y is an element of R^k.

Interpret this geometrically, as a statement about parallelograms.
• Jun 12th 2006, 01:39 PM
ThePerfectHacker
Quote:

Originally Posted by Susie38
I don't know if this the right thread to get help, but here it goes.

Prove that:

|x + y|^2 + |x-y|^2 = 2|x|^2 + 2|y|^2 if x is an element of R^k and y is an element of R^k.

Interpret this geometrically, as a statement about parallelograms.

Considering $\displaystyle \mathbb{R}^k$ as a vector space over $\displaystyle \mathbb{R}$.
Then,
$\displaystyle x=(a_1,a_2,...,a_k)$
$\displaystyle y=(b_1,b_2,...,b_k)$
Then,
$\displaystyle x+y=(a_1+b_1,a_2+b_2,...,a_k+b_k)$
Then,
$\displaystyle |x+y|=\sqrt{(a_1+b_1)^2+...+(a_k+b_k)^2}$
Thus,
$\displaystyle |x+y|^2=(a_1+b_1)^2+...+(a_k+b_k)^2$
Similarily,
$\displaystyle |x-y|^2=(a_1-b_1)^2+...+(a_k-b_k)^2$
When you add them together you have,
$\displaystyle \sum_{i=1}^k(a_i^2+2a_ib_i+b_i^2)+\sum_{i=1}^k(a_i ^2-2a_ib_i+b_i^2)$=$\displaystyle \sum_{i=1}^k2a_i^2+2b_i^2=2|x|^2+2|y|^2$