Under what conditions will these sets be orthogonal sets?

• Mar 11th 2010, 05:34 AM
Runty
Under what conditions will these sets be orthogonal sets?
Suppose that $\displaystyle \{u,v,w\}$ is an orthogonal set. Under what conditions will the set $\displaystyle S_1=\{u+v,u-v,w\}$ be an orthogonal set?

Repeat for $\displaystyle S_2=\{u+v-w,u-v+w,v+w\}$ and $\displaystyle S_3=\{u-v,u+v,w-u,u+v+w\}$.
• Mar 11th 2010, 05:48 AM
josipive
Quote:

Originally Posted by Runty
Suppose that $\displaystyle \{u,v,w\}$ is an orthogonal set. Under what conditions will the set $\displaystyle S_1=\{u+v,u-v,w\}$ be an orthogonal set?

Repeat for $\displaystyle S_2=\{u+v-w,u-v+w,v+w\}$ and $\displaystyle S_3=\{u-v,u+v,w-u,u+v+w\}$.

vectors x and y are orthogonal if: (x|y) = 0 ( scalar product )

if set { u,v,w } is orthogonal that means that:

( u|v ) = 0 and ( u|w ) = 0 and ( v|w ) = 0

what conditions do you need for set { u+v, u-v,w } to be orthogonal?

it must be:

( u + v | u - v ) = 0 and ( u + v | w ) = 0 and ( u - v | w ) = 0

1. ( u + v | u - v ) = ( u|u ) + ( v|u ) + ( u|v ) - ( v|v ) = ( u|u ) - ( v|v ) ---- so first condition is:

( u|u ) = ( v|v )

2. it is ok ( u + v | w ) = ( u|w ) + ( v|w ) = 0
3. it is ok

so condition is ( u|u ) = ( v|v )

S_2 and S_3 in the same way
• Mar 11th 2010, 08:30 AM
Runty
I've gone through all of it, and I'm somewhat concerned about the answer for $\displaystyle S_3$. I end up getting $\displaystyle u\cdot u=v\cdot v,u\cdot u=w\cdot w,u\cdot u=0$, so $\displaystyle v\cdot v=w\cdot w=0$. This would imply (I think) that $\displaystyle S_3$ is always orthogonal. Is this true, or did I do something wrong?