Value of combinaison of orthogonal elements

[Dassinia]

Hello this is not a homework, just studying for the exam, and :

Homework Statement

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||

Homework Equations

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 ?

 

[Office_Shredder]

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

 

[Dassinia]

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