Does vector projection distribute over addition?

[Poobel]

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.

 

[Galileo]

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.

 

[Poobel]

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?

 

[Galileo]

Just give a counter example. You'll have to look for one. That's sufficient.

 

[Poobel]

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?

 

[Galileo]

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.

 

[Poobel]

OK thanks a bunch