Triangle Inequality: Prove ||a-b|| for R^n

[andrassy]

Homework Statement

Using the triangle inequality show that for all vectors a, b, c in R^n:

||a-b|| <= ||a - c || + ||c - b ||

(a, b, c are vectors)

Homework Equations

The triangle inequality states:
||a+b|| <= ||a|| + ||b||

The Attempt at a Solution

Im really not sure what to do he made our first problem set due before we finished learning the information. I tried to do something with ||c|| = || sqrt (b^2 - a^c) || (by pythagorean theorem) but i wasn't able to come up with anything. Any guidance in the right direction would be helpful

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Also, we are given parametric equations in R^3 and have to determine if the lines are parallel, intersect, or are skew. I can't seeem to find this in the book. Any tips or hints on the general procedure?

Thanks!

 

[danago]

Isnt that result just the triangle inequality itself with the 'a' replaced by a-c and the b replaced with the c-b, so that the left hand side ends up being (a-c) + (c-b) = a-b?

 

[andrassy]

Yeah i guess it is but can you do that? How is a = a - c

Oh if you make b = a - c and a = c - b it works and if its a triangle that should be true. THanks!