Spivaks Calculus on a Manifold

[Storm Butler]

Homework Statement

Prove the triangle inequality theorem $$\leftX-Y\right\rfloor\lfloor$$$$\leq\leftZ-Y\right\rfloor\lfloor$$+$$\leftY-X\right\rfloor\lfloor$$. (my computer just shows the latex as a bunch of script so i don't know if that came out right.

Homework Equations

The Attempt at a Solution

So this isn't actually for H.W. i was just trying to work through the examples in spivaks book. That being said i don't really know if there is an appropriate way to approach this or anything but i thought i might be able to get it by looking at the Law of cosines and trying to se if i could get it to look anything like it and go on from there. However, it isn't obvious to me if this is a good path to take or not. So i would just like someone to kind of give me a hint and tell me if what I am attempting is right or a hint towards another way that might be better. (please don't give the answer though as i would like to work out as much of it as i can myself).
-Thanks

 

[mighty2000]

Hi there, it's great that you're trying to work through the examples in Spivak's book!

To prove the triangle inequality theorem, there are a few different approaches you could take. One approach would be to use the properties of the real numbers and basic algebra to manipulate the given expression into a form that is more easily recognizable as the triangle inequality. Another approach would be to use the properties of triangles and basic geometry to show that the given expression is a valid representation of the triangle inequality.

Since you mentioned the Law of Cosines, let's explore that approach a bit more. The Law of Cosines states that for a triangle with sides of lengths a, b, and c, and angles A, B, and C opposite those sides, we have c^2 = a^2 + b^2 - 2ab*cos(C). This can also be written as c^2 = (a-b)^2 + 2ab*(1-cos(C)). Notice that this is similar to the given expression, except for the absolute value signs.

So, one approach could be to try to manipulate the given expression to look more like the Law of Cosines. Can you think of any properties of absolute values or algebraic manipulations that could help you do this? Once you have the expression in a form that resembles the Law of Cosines, you could then use the properties of triangles to show that it is equivalent to the triangle inequality.

I hope this helps and gives you a good starting point. Good luck with your problem solving!