How Do Orthogonal Vectors Determine Unique Scalar Coefficients?

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Homework Help Overview

The discussion revolves around the concept of orthogonal vectors and their role in expressing an arbitrary vector A as a linear combination of three nonzero orthogonal vectors F, G, and H. Participants are exploring the uniqueness of the scalar coefficients x, y, and z in the equation A = xF + yG + zH.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the use of the dot product to derive the scalars x, y, and z. Some express difficulty in visualizing the problem and applying the dot product effectively. Questions about the meaning of orthogonality and its implications for the uniqueness of the scalars are raised.

Discussion Status

The discussion is ongoing, with some participants suggesting the use of the dot product as a method to find the scalars. There is a recognition of the need for clarity on the concept of orthogonality and its relevance to the problem, but no consensus has been reached on a solution approach.

Contextual Notes

Participants are working under the constraints of a homework assignment, which may limit the information they can access or the methods they can employ. There is an emphasis on understanding the process rather than receiving direct solutions.

nepenthe
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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..
 
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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.
 
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..
 
nepenthe said:
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.
 
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??
 
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?
 

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