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If two vectors v, w are both unit vectors, then v+w and v-w will be orthogonal, but are v+w and v-w also unit vectors?

I would say no because the inner product of the two added, and the two subtracted would also have to be orthogonal.

<(v+w)+(v-w),(v+w)-(v-w)>

= <2v,2w>

and <2v,2w> must be greater than zero for the product to be defined in the first place.

*EDIT:

If it helps, the way I originally showed that they were orthogonal was to take

<v+w,v-w> = ||v||^2 - ||w||^2

if they are unit vectors then ||v|| = 1 and ||w|| = 1 so

<v+w,v-w> = 1 - 1 = 0 = orthogonal

I would say no because the inner product of the two added, and the two subtracted would also have to be orthogonal.

<(v+w)+(v-w),(v+w)-(v-w)>

= <2v,2w>

and <2v,2w> must be greater than zero for the product to be defined in the first place.

*EDIT:

If it helps, the way I originally showed that they were orthogonal was to take

<v+w,v-w> = ||v||^2 - ||w||^2

if they are unit vectors then ||v|| = 1 and ||w|| = 1 so

<v+w,v-w> = 1 - 1 = 0 = orthogonal

Last edited: