find a vector in an orthnormal basis

suppose I have 3 vectors, a b and c that form an orthonormal basis. How would I find a vector v = (1,2,3) in the a b c system? I don't really understand what its asking. Do I have to find a linear combination from a b c that forms v?

so basically for some scalars x y and z, I have to find

xa + by + cz = v?

if I'm right, how would I go about doing that?

Re: find a vector in an orthnormal basis

Re: find a vector in an orthnormal basis

wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?

Re: find a vector in an orthnormal basis

Quote:

Originally Posted by

**Kuma** wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?

The angle brackets mean inner product, i.e. . Make sense now?

Re: find a vector in an orthnormal basis

ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.

Re: find a vector in an orthnormal basis

Quote:

Originally Posted by

**Kuma** ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.

That works too haha