Vector triangle inequality and when equality holds

If we follow one vector and then another, the sum connects our starting point to our ending point. The direct route can be shorter than the two parts of the trip. The vector triangle inequality expresses that comparison using lengths.

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Compare the two ways of measuring the trip

$\Vert u+v\Vert \le \Vert u\Vert +\Vert v\Vert$

The diagonal is no longer than the total distance along the other two sides.

Equality holds when the two nonzero vectors point in the same direction, or when at least one vector is zero. For a pair pointing in different directions, the direct route is shorter.

Work through the length calculation

Let $u=(2,-1)$ and $v=(-1,4)$. Add corresponding components to find $u+v=(1,3)$.

$\begin{array}{l}\Vert u+v\Vert =\sqrt{1^2+3^2}=\sqrt{10}\\ \Vert u\Vert +\Vert v\Vert =\sqrt{5}+\sqrt{17}\end{array}$

The lengths are approximately $3.162$ and $6.359$, so the inequality is strict for this pair.

Check direction when the vectors are collinear

For $u=(3,4)$ and $v=2u=(6,8)$, the sum has length $15$, which equals $5+10$. If instead $v=-2u=(-6,-8)$, then $u+v=-u$ has length $5$, which is less than $15$. This pair is still dependent because $v$ is a scalar multiple of $u$.

To test the independence of a pair from dot products and lengths, use the equality condition in Cauchy–Schwarz.

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