# Triangle Inequality and Cauchy Inequality Proofs

• jumbogala
In summary, the student is having trouble visualizing the proofs for Cauchy's Inequality and Triangle Inequality. The first proof is that |A dot B| / (|A| dot |B|) = cos(x), and cos(x) ≤ 1. The second proof is that |A • B | ≤ |A| • |B|. The student is having trouble seeing how to geometrically prove these inequalities, and is stuck using algebraic proofs. However, the algebraic proofs are the same for both inequalities. The student is also having trouble understanding how to find the projection of a vector onto another vector. This problem is solved by explaining that |A dot B| and |A •
jumbogala

## Homework Statement

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.

## 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.

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

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 into the same problem with the other proof.

| 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.

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?

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.

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?

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

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?

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?

A vector with length 15?

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

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?

A groups of B things?

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

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!

## What is the Triangle Inequality Proof?

The Triangle Inequality Proof states that for any triangle with sides A, B, and C, the sum of any two sides must be greater than the third side. In mathematical notation, this can be written as A + B > C, B + C > A, and A + C > B.

## What is the Cauchy Inequality Proof?

The Cauchy Inequality Proof, also known as the Cauchy-Schwarz Inequality, states that for any two vectors A and B, the absolute value of their dot product is less than or equal to the product of their magnitudes. In mathematical notation, this can be written as |A ∙ B| ≤ |A||B|.

## What is the significance of these proofs?

The Triangle Inequality and Cauchy Inequality Proofs are important in various mathematical disciplines, such as geometry, linear algebra, and analysis. They serve as fundamental principles that can be used to prove more complex theorems and solve problems in these areas.

## What are some real-world applications of these proofs?

The Triangle Inequality Proof can be applied in various fields such as engineering, physics, and economics to determine the maximum possible distance between two points or to optimize the use of resources. The Cauchy Inequality Proof is commonly used in statistics to measure the correlation between variables and in signal processing to analyze signals.

## What are some common mistakes or misconceptions when applying these proofs?

One common mistake is assuming that the Triangle Inequality Proof only applies to triangles when in fact it can be extended to any polygon with straight sides. Another misconception is applying the Cauchy Inequality Proof to non-vectors, as it only applies to vectors in a vector space. It is also important to note that both proofs have strict conditions and may not hold true in all cases.

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