How Do Orthogonal Vectors Determine Unique Scalar Coefficients?

[nepenthe]

orthogonal vectors

Hello..Can someone help me with the following question?

Let F,G, and Z be nonzero vectors, each orthogonal to other two..Let A be any vector.Show that there are unique scalars x,y, and z such that A=xF+yG+zH.

Thank You..

 

[Physics Monkey]

You want to write an arbitrary vector A in terms F, G, and H as A=xF+yG+zH.
If you can find a formula for x,y,z in terms of A, F, G, and H then you have solved your problem, right? Hint: think about the dot product.

 

[nepenthe]

hello..I thought about using dot product.But I couldn't apply to this problem..and I think the main problem is that I can't imagine this question in my head..
could you please give me the solution(just to understand the the process)..and also solution methods for these "show that or prove that" questions..what is the first thing that I have to think?

Thankk you..

 

[HallsofIvy]

hello..I thought about using dot product.But I couldn't apply to this problem..and I think the main problem is that I can't imagine this question in my head..
could you please give me the solution(just to understand the the process)..and also solution methods for these "show that or prove that" questions..what is the first thing that I have to think?
Thankk you..

You "thought" about using the dot product? Why not just use it?

Do you know what "orthogonal" means? I suggest you look it up.

 

[nepenthe]

First, because I couldn't solve this ploblem, I am here..

I know what orthogonal means..I wrote equations for dot product..But I have no idea how to show there are unique scalars? or why there are unique scalars x,y, and z??

 

[HallsofIvy]

DO IT!
The best way to prove something exists is to show how to find it.
If A= xF+yG+zH, what is the dot product of both sides with F? Can you solve that equation for x?