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, \begin{eqnarray*} \norm{\vec{v}+\vec{w}}^2 & = & (\vec{v}+\vec{w})\dotp (\vec{v}+\vec{w})\\ &=& \vec{v}\dotp \vec{v}+\vec{v}\dotp \vec{w}+\vec{w}\dotp \vec{v}+\vec{w}\dotp \vec{w}\\ &=&\norm{\vec{v}}^2+ 2(\vec{v}\dotp \vec{w})+\norm{\vec{w}}^2 \end{eqnarray*}
- 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. \begin{eqnarray*} \norm{\vec{v}+\vec{w}}^2 &=& \norm{\vec{v}}^2+2(\vec{v}\dotp \vec{w})+\norm{\vec{w}}^2\\ &\leq & \norm{\vec{v}}^2+2\norm{\vec{v}}\norm{\vec{w}}+\norm{\vec{w}}^2\\ &=&\left (\norm{\vec{v}}+\norm{\vec{w}}\right )^2 \end{eqnarray*}Taking the square root of both sides yields the desired result.