Dot Product and Perpendicularity

[imsoconfused]

Homework Statement

If |A+B|^2=|A|^2+|B|^2, prove that A is perpendicular to B.

Homework Equations

A^2=|A||A|

The Attempt at a Solution

All I can think of to do is expand the equation to get (A.B)(A.B)=A.A+B.B. I know that iff a.b=0, the two are perpendicular but I can't figure out how to show that because I don't really even understand what the question is asking. I'm in desperate need of some help!

 

[Defennder]

As said before, start with the LHS, not the RHS. Your approach uses the RHS. What is another way of writing |A+B|^2 using the dot product?

 

[imsoconfused]

Does it involve taking the square root of both sides?

 

[Defennder]

Nope, not at all.

 

[imsoconfused]

Yes, I just tried it and it was definitely the wrong idea. When you say I need to begin with the LHS, though, do you mean I need to write it out like (A.B)(A.B) to obtain 2AB? If that's not it, could you give me a hint to get me going in the right direction?

 

[imsoconfused]

Wait, 2AB is the wrong thing, I looked at the something else I'd written down that looked similar. The dot product there would be 2A+2B, right?

 

[Defennder]

Well I gave you a hint in the other thread. This is the third thread with the same question. You ought to stick to one thread and certainly only one forum. $$|A|^2 = \mathbf{A} \cdot \mathbf{A}$$

 

[Ben Niehoff]

|A+B|^2 is NOT equal to (A.B)^2.

Try letting C = A+B. Then what is |C|^2?

 

[imsoconfused]

oh, whoops, I messed that last thing up entirely. I'm way too tired, this prof gave way too much homework tonight. I didn't think to use a third variable, that seems like a good idea. defennder, you said to begin with the LHS so I thought that |A|^2=A.A wouldn't yet be relevant.


also, defennder: someone moved my first post to a different forum, so I couldn't find it and reposted it in another one. then I had so many questions in the thread that I thought it'd be less intimidating and I'd get more answers if I asked just the one question. sorry! patience is a virtue god failed to give me. =)

 

[imsoconfused]

I keep coming back to the fact that 0=2AB. I'm not sure if this has any significance, but if I multiply out |A+B|^2 I get A^2+2AB+B^2, and if I set that equal to the RHS and simplify, I get 2AB=0.

 

[imsoconfused]

no, that seems like it would prove the two are NOT perpendicular. why is this so hard for me?

 

[imsoconfused]

actually, it was the right train of thought, am I right that 2(A.B)=0 is significant?

 

[Ben Niehoff]

Yes. What does it mean if the dot product between two vectors is zero?

 

[imsoconfused]

that they are perpendicular! I think I've finally got a viable proof--thank you both so much for your help! now it is off to bed for a couple hours before class starts... good night!