Proving Orthogonality of Vectors Using Linear Algebra Techniques

[Yosty22]

Homework Statement

Show that B|A|+A|B| and A|B|-B|A| are orthogonal.

Homework Equations

N/A

The Attempt at a Solution

I'm not too sure exactly how to start this. I do know that for two things to be orthogonal, the dot product has to be equal to 0, but I'm not sure how to evaluate this at all.

 

[vela]

What specifically is stopping you from taking the dot product of the two?

 

[Yosty22]

I guess my main issue right now is knowing what exactly is meant by |A| and |B|? Does this mean I can create an arbitrary matrix A and matrix B and replace |B| and |A| with the determinant of the two matrices? So if I have an arbitrary 2x2 matrix |a1 a2| and |b1 b2| I can calculate the determinant, then do the dot product?
|a3 a4| |b3 b4|

 

[vela]

I would think that |A| denotes the length of vector A, not the determinant of a matrix A. In any case, you don't need to calculate that part out. You just need to know that |A| and |B| are scalars.

 

[Yosty22]

Okay, I understand they are scalars, but I'm not quite sure where to go from here. Do I need to do something with the Commutator?

 

[vela]

No. Why would there be a commutator involved? I think you're confusing matrix multiplication with the dot product. They're not the same operation.

The result you're being asked to show is true for any vector space and appropriate dot product. You really don't need to know anything about the specifics of how to calculate the dot product. You just need to know properties of the dot product in general.

 

[Yosty22]

Ahh, I see. What if I did this:

<B|A|+A|B|> dot <A|B|-B|A|>

If you solve the dot product there, you get B|A|*A|B|-A|B|*B|A|. You have the same thing on either side of the subtraction sign, so it has to be 0. If the dot product is 0, then the vectors are orthogonal. Would that work?

 

[vela]

What happened to the other two terms?

 

[Yosty22]

Oh whoops, I had them written down on the paper, I just forgot to type it. So in total, you have:

(B|A|)*(A|B|) - (B|A|)*(B|A|) + (A|B|)*(A|B|) - (A|B|)*(B|A|).

From this, everything cancels down to 0, so the vectors have to be orthogonal because the dot product is zero. Right?

 

[vela]

Yup, you got it.

 

[LCKurtz]

Oh whoops, I had them written down on the paper, I just forgot to type it. So in total, you have:

(B|A|)*(A|B|) - (B|A|)*(B|A|) + (A|B|)*(A|B|) - (A|B|)*(B|A|).

From this, everything cancels down to 0, so the vectors have to be orthogonal because the dot product is zero. Right?

Yup, you got it.

I don't quite agree. I would want to see a bit more simplification before I agree that he's got it. Like why the middle two terms cancel. I'm not convinced he understands yet.

 

[vela]

Fair enough.