# Value of combinaison of orthogonal elements

1. Oct 23, 2013

### Dassinia

Hello this is not a homework, just studying for the exam, and :
1. The problem statement, all variables and given/known data
Consider E a linear space with dot product (.,.) and the norm ||x|| = sqrt(x,x)
a and b two orthogonals elements of E
Find the value of ||a+ b|| et||a- b|| and ||a+ b||-||a- b||

2. Relevant equations

3. The attempt at a solution
a and b orthogonals means that ||a||=||b||=1
||a+b||=√(a+b,a+b) = √[(a,a)+(b,a)+(a,b)+(b,b) ] =√[ 1+0+0+1 ] = √2
||a-b||=√(a-b,a-b) = √[(a,a)+(-b,a)+(a,-b)+(-b,-b) ] =√[ 1-(b,a)+(-b,a)*-1 ] = √[ 1-(b,a)-(b,a)*-1 ]=0
||a+b||-||a-b||=√2

Is that correct ?

2. Oct 23, 2013

### Office_Shredder

Staff Emeritus
If ||b|| = 1, then (-b,-b) = 1, not -1.

Also orthogonal typically just means that (a,b) = 0, without giving any restriction on the size of the vector (if they are all unit vectors they're typically called orthonormal), so you should probably double check the wording of the problem/your class's definition of orthogonal

3. Oct 24, 2013

### Dassinia

Hello,
They're orthonormal, I just made a mistake copying the problem statement.
So
||a-b||=√2
||a+b||-||a-b||=0