Solving Audio Stereo Problem: Expression of (M^2) + (D^2)

[Xkaliber]

Homework Statement

The problem is given in the attachment below. The relevant information is provided under the "Stereo Audio" heading. Additionally, I have a couple of pieces of extra information not listed in the Stereo Audio section:

M is the component of (column vector) [R, L] on (column vector) [1, 1]
(This notation means that R and L are two entries in a column vector. The "R" being the upper entry and the "L" being the lower entry. The same applies for the 1 and 1)

D is the component of (column vector) [R, L] on (column vector) [-1, 1]


For the first problem, it asks me to express (M^2) + (D^2) in terms of L and R. Am I correct in writing (M^2) + (D^2) = [(L+R)^2]/4 +[(L-R)^2]/4?

 

[Xkaliber]

Is there a reason that my attachment has not been approved? How long does it usually take?

 

[Xkaliber]

Well, I guess I can just post a picture instead. Click the thumbnail image to enlarge.

http://img166.imageshack.us/img166/2990/clipboard01tu5.th.jpg

 

[HallsofIvy]

It takes a while for any link to another site to be approved. (Some brave soul with really really good virus protection has to check it out.)

I don't see that there is really any "Linear Algebra" involved here.

Yes, if M= (L+ R)/2 and D= (L-R)/2 then M²+ D²= (L+R)²/4+ (L-R)²/4. It would be a good idea, I think, to multiply those squares out and combine the fractions.

 

[Xkaliber]

Yeah, that's what it seemed to me as well.

Additionally, I can't seem to be able to picture exactly what this scenario would look like graphed. First, we are given M= (L+ R)/2, which appears to be half the length of the vector [R, L]. However, at the same time I am told M is the component of vector [R, L] on [1, 1] (basically, a projection of [R, L] onto the [1, 1] vector) However, it seems these cannot be true at the same time, at least in my mind. So is it possible to graph these based on the given information?

 

[HallsofIvy]

Yeah, that's what it seemed to me as well.

Additionally, I can't seem to be able to picture exactly what this scenario would look like graphed. First, we are given M= (L+ R)/2, which appears to be half the length of the vector [R, L]. However, at the same time I am told M is the component of vector [R, L] on [1, 1] (basically, a projection of [R, L] onto the [1, 1] vector) However, it seems these cannot be true at the same time, at least in my mind. So is it possible to graph these based on the given information?

No, (L+ R)/2 is certainly not "half the length of the vector [R,L]"! That would be (1/2)\sqrt{L^2+ R^2}.
Drawing a right triangle diagram, you should be able to see that the projection of vector u on vector v has length |u|sin\theta where \theta is the angle between u and v. Since u.v= |u||v|sin\theta, we can write that as u.v/|v|. In the particular case that u= [L, R] and v= [1, 1] that is (L+ V)/\sqrt{2}.

 

[Xkaliber]

So is M= (L+ R)/2 a sum of two vectors or the sum of their magnitudes?