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-   -   Triangle Inequality and Cauchy Inequality Proofs (http://www.physicsforums.com/showthread.php?t=434567)

 jumbogala Oct3-10 06:22 PM

Triangle Inequality and Cauchy Inequality Proofs

1. The problem statement, all variables and given/known data
The question says to find a proof for Cauchy's Inequality and then the Triangle Inequality.

This is an elementary linear algebra class I'm doing, so I can't use inner products or anything.
2. Relevant equations

3. The attempt at a solution
I got the proofs using algebra, but I'm having trouble seeing them geometrically. I want to draw a picture for Cauchy's Inequality but can't quite visualize it.

 Inferior89 Oct3-10 06:29 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

Edit: Misread the question. Seems like you are supposed to find both the algebraic and geometric proof for both the inequalities.

 jumbogala Oct3-10 06:42 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

Yeah, that's the way I read it too.

But my proof for the first one is just that | A dot B| / (|A| dot |B|) = cos(x), and cos(x) ≤ 1, so |A dot B| ≤ |A| dot |B|.

Given the definitions we got for this class, I don't see any way to prove this differently, unless I use inner products. I run in to the same problem with the other proof.

 JonF Oct3-10 06:51 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

| A • B | ≤ |A| |B|
Algebraic: you’re right since cos is bound between 0 and 1 this shows this inequality must be true.
Geometric: Think about what the dot product represents on a graph. It’s magnitudes of one vector projected onto another.

| A + B| ≤ |A| + |B|
Algebriac: Hint: Add |-a| <= a <= |a| and |-b| <= b <= |b|
Geometric: This just means that no one leg of a triangle can be longer than the other two put together.

 jumbogala Oct3-10 07:13 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

For the geometric proof of | A • B | ≤ |A| |B|,

I can use that Bcos(x) = (B dot A) / |A| = B dot n, where n is a unit vector in the direction of A. That's a projection.

By definition n = A / |A|, so B dot (A / |A|) = Bcos(x) --> B dot A = ABcos(x), but that just brings me back to the proof I did first, so is it really any different?

 JonF Oct3-10 07:25 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

IMHO that's more of an algebraic proof. In this context I think geometric means proof by picture rather than from the axioms of geometry.

 jumbogala Oct3-10 08:33 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

I guess I'm having trouble seeing how you can prove it using a picture.

I was trying to use a picture similar to the one on http://en.wikipedia.org/wiki/File:Dot_Product.svg

But since |A dot B| and |A||B| are numbers, how can you show them on the diagram?

 JonF Oct3-10 08:54 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

What geometric shape would be represented |A||B|?
What is geometricly represented by that picture.

 jumbogala Oct3-10 09:05 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

The picture geometrically represents the projection of A onto B, and that projection is A dot B. Is that right?

But |A||B| is a number, how can it have a geometric interpretation?

 JonF Oct3-10 09:07 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

Don’t over think it, If I asked you to draw a picture of 3*5, or maybe build it with blocks, how would you do it?

 jumbogala Oct3-10 09:09 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

A vector with length 15?

But how would you know which direction the vector would point?

 JonF Oct3-10 09:11 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

Don't think of |A|*|B| in terms of vectors, |A|*|B| is just some number, |A| is just some number, |B| is just some number. Who cares where we got them from.

How do they teach little kids to understand multiplication? They always compare it to some shape.

Now ask your self could the projection of A onto B ever be larger than that shape?

 jumbogala Oct3-10 09:20 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

A groups of B things?

 JonF Oct3-10 09:22 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

3*5 is often represented as a rectangle of height 3 length 5.

 jumbogala Oct3-10 10:16 PM

Re: Triangle Inequality and Cauchy Inequality Proofs

Ah, I see it now. The rectangle describing |A dot B| will always have a smaller width than the other one, because the projection of B on A is smaller than B itself.

Unless the angle is zero, in which case they're equal. Thanks for all your help!

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