Vector Projection Proof: Does aproj.(b+c) = aproj.b + aproj.c?

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Homework Help Overview

The discussion revolves around the proof of a vector projection identity involving three vectors, a, b, and c. The original poster questions whether the projection of the sum of two vectors (b+c) onto a vector a equals the sum of the individual projections of b and c onto a.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the validity of the proposed vector projection equality and discuss the implications of vector direction versus magnitude. There are suggestions to find counterexamples to demonstrate the inequality.

Discussion Status

Some participants have provided guidance on seeking counterexamples, noting that the direction of the vectors is crucial to the proof. There is an acknowledgment that the original proposition may not hold true in general, and multiple interpretations of the problem are being considered.

Contextual Notes

Participants are discussing the implications of linear dependence among the vectors b and c and how that affects the projection results. There is an emphasis on finding specific examples to illustrate the points being made.

Poobel
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Me again.

I would really appreciate if you could help me with the following proof:

a,b,c are vectors

Does aproj.(b+c) =aproj.b + aproj.c

Sorry for notation.

Thank you.
 
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You won't be able to prove it, since the result is not true in general.

Notice that only the direction of the vector whose span you are projecting on is important, not its magnitude. Use this to find a simple counterexample.
 
Umm yeah the question asks does blah=blah. I thought that it would not be true, however I am clueless how to put it on paper that the direction would not be the same. Should I call sum of vectors b and c a vector d?
 
Just give a counter example. You'll have to look for one. That's sufficient.
 
Could you be a little more specific? Or should I just do right and left side for 3 random vectors, and thus prove that the equality is not true?
 
Yes.

Generally, the easier the counterexample the better. Since the reason for the deficit is known it can help you to find an example:
Take b and c linearly dependent. Then projecting on b is the same as projecting on c or projecting on b+c. So the right side is twice the left side.
 
OK thanks a bunch
 

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