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