Triangle Inequality: Proving and Understanding

In summary: Next, prove that the triangle inequality is an equality if and only if |a>=alpha|b> where alpha is a real positive scalar. This can be done by first proving that the triangle inequality is an equality if and only if |a>=|b|. To do this, use the cauchy-schwartz inequality to show that (a,b) = |a||b|. Next, use the fact that ||a|| = SQRT (a a*), to show that (a,b) = |a||b|. Finally, use the triangle inequality to conclude that the triangle inequality is an equality.
  • #1
blanik
15
0
I have proven the triangle inequality starting with ||a+b||^2 and using the Schwartz Inequality. However, the next part of the problem says:

"Show that the Triangle Inequality is an equality if and only if |a>=alpha|b> where alpha is a real positive scalar." It must be proved in both directions.

Any help on where to begin would be greatly appreciated.
 
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  • #2
blanik said:
I have proven the triangle inequality starting with ||a+b||^2 and using the Schwartz Inequality. However, the next part of the problem says:

"Show that the Triangle Inequality is an equality if and only if |a>=alpha|b> where alpha is a real positive scalar." It must be proved in both directions.

Any help on where to begin would be greatly appreciated.

begin by proving one direction. complete the proof by proving the other direction. I'm not sure what |a>=alpha|b> means, is there another way to explain what that says?
 
  • #3
|a>=alpha|b> means the vector A equals alpha times the vector B where alpha is a real positive scalar. Does that help?

I understand that I am "supposed" to start with one way and go the other, but what does that mean? Do I substitute a=alpha b for a and solve for ||alpha b + b|| = ||alpha b|| + ||b||? I have been playing around with the definition of ||a|| = SQRT (a a*), etc...
 
  • #4
in your proof of the triangle inequality make all your inequalities equalities & see what you get. at the step where you use the cauchy-schwartz inequality you see that (a,b) = |a||b|. if one vector is a multiple of the other then figure out that (a,b) = |a||b| is true. for the other way suppose that's true. then by the cauchy-schwartz inequality one vector is a multiple of the other. ( ( , ) means inner product & | | means length)
 
  • #5
The triangle inequality becomes an equality when the Schawrtz inequality becomes an equality. Read through the proof of the Schwartz inequality to see when this happens.
 
  • #6
Start by explicitly writing out what [tex] ||~ |a \rangle + |b \rangle ~ ||^2 [/tex] is. You might start seeing where the Scwartz inequality comes into play.
 

FAQ: Triangle Inequality: Proving and Understanding

What is the Triangle Inequality?

The Triangle Inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In other words, in a triangle ABC, AB + BC > AC, AC + BC > AB, and AB + AC > BC.

Why is the Triangle Inequality important?

The Triangle Inequality is important because it is a fundamental property of triangles and is used to prove many geometric theorems. It also helps us determine if a given set of lengths can form a triangle.

How can the Triangle Inequality be proved?

The Triangle Inequality can be proved using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. By manipulating the equation, we can show that the sum of the lengths of any two sides must be greater than the length of the third side.

Can the Triangle Inequality be applied to all types of triangles?

Yes, the Triangle Inequality applies to all types of triangles, including equilateral, isosceles, and scalene triangles. It is a universal property of triangles.

Are there any other properties of triangles related to the Triangle Inequality?

Yes, there are several other properties of triangles that are related to the Triangle Inequality, such as the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle is always 180 degrees. The Triangle Inequality also helps us determine the shortest distance between two points, known as the straight line distance.

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