Triangle Inequality
The Triangle Inequality is a simple, yet powerful result used widely in analysis and topology as well as other branches of mathematics. The triangle inequality has its roots in geometry. It initially appeared as a proposition in the Elements - a treatise comprised of thirteen books covering plane and solid geometry, and number theory - written by Euclid of Alexandria around 300 B.C.
The geometric version of the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length the third side.
Here we are interested in the vector version of this result. Given vectors and , we have
Intuitively, we observe that equality occurs when either or (or both) are zero, or when non-zero vectors and point in the same direction, otherwise the inequality is strict.
Proving the triangle inequality requires some preliminary results.
- Proof
- By Theorem th:dotproductproperties we have,
- Proof
- Recall that by Theorem th:dotproductcosine, , where is the included angle. Our result follows from the fact that .
- Proof
- We will use Lemma lem:triLem and Theorem th:CS.
Taking the square root of both sides yields the desired result.