Orthogonal set - Geometric interpretation

[shounakbhatta]

Orthogonal set -- Geometric interpretation

Hello,

If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0.

Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero.

Secondly, if I have three vectors, v1,v2,v3 with certain values, computing, we get <v1,v2>=0
<v1,v3>=0
<v2,v3>=0

How will it look like in geometrical figure. I mean to say, what if we take these 3 vectors and put it on a picture and how it would look like?

Thanks,

-- Shounak

 

[Erland]

Hello,

If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0.

Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero.

Secondly, if I have three vectors, v1,v2,v3 with certain values, computing, we get <v1,v2>=0
<v1,v3>=0
<v2,v3>=0

How will it look like in geometrical figure. I mean to say, what if we take these 3 vectors and put it on a picture and how it would look like?

Thanks,

-- Shounak

In R³, these will be three perpendicular vectors. For example, the standard basis vectors (1,0,0), (0,1,0), and (0,0,1).
In R², this is not possible, unless at least one of the vectors is the zero vector.