C.'s question at Yahoo Answers (orthogonality).

[Fernando Revilla]

Here is the question:

Show that vectors X and Y in R^n are orthogonal if and only if
||X+Y|| = ||X-Y||

Here is a link to the question:

Can someone help me with this linear algebra question (Orthogonality) ? - Yahoo! Answers

I have posted a link there to this topic so the OP may find my response.

 

[Fernando Revilla]

Hello C.

Using the definition of norm:

$ \left\|x+y\right\|^2=(x+y)\cdot (x+y)=x\cdot x+x\cdot y+y\cdot x+y\cdot y= \left\|x\right\|^2+ \left\|y\right\|^2+2\;x\cdot y\\
\left\|x-y\right\|^2=(x-y)\cdot (x-y)=x\cdot x-x\cdot y-y\cdot x+y\cdot y= \left\|x\right\|^2+ \left\|y\right\|^2-2\;x\cdot y\\
$
If $x$ and $y$ are orthogonal, then $x\cdot y=0$ as a consequence $\left\|x+y\right\|^2=\left\|x-y\right\|^2$ or equivalently $\left\|x+y\right\|=\left\|x-y\right\|$.

On the other hand if $\left\|x+y\right\|=\left\|x-y\right\|$, then $\left\|x+y\right\|^2=\left\|x-y\right\|^2$ which implies $4\;x\cdot y=0$ or equivalently $x\cdot y=0$ that is, $x$ and $y$ are orthogonal.