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 \(\vec {v}\) and \(\vec {w}\), we have

\[\norm {\vec {v}+\vec {w}}\leq \norm {\vec {v}}+\norm {\vec {w}}\]

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Intuitively, we observe that equality occurs when either \(\vec {v}\) or \(\vec {w}\) (or both) are zero, or when non-zero vectors \(\vec {v}\) and \(\vec {w}\) point in the same direction, otherwise the inequality is strict.

Proving the triangle inequality requires some preliminary results.

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*}

Recall that by Theorem th:dotproductcosine, \(\vec {v}\dotp \vec {w}=\norm {\vec {v}}\norm {\vec {w}}\cos \theta \), where \(\theta \) is the included angle. Our result follows from the fact that \(|\cos \theta |\leq 1\).

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.