We can write

$\stackrel{?}{\left|OM\right|}$ = $\left|\stackrel{?}{a}\right|\dots \dots \dots .\left(\mathrm{i}\right)$

$$ $\stackrel{?}{\left|MN\right|}$ = $\left|\stackrel{?}{b}\right|$ = $\stackrel{?}{\left|OP\right|}$ ………(ii)

$$ $\stackrel{?}{\left|ON\right|}$ = $\left|\stackrel{?}{a}+\stackrel{?}{b}\right|\dots \dots .\left(iii\right)$

In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON < (OM + MN)

$\left|\stackrel{?}{a}+\stackrel{?}{b}\right|$ < $\left|\stackrel{?}{a}\right|$ + $\left|\stackrel{?}{b}\right|$ …..(iv)

If the two vectors $\stackrel{?}{a}$ and $\stackrel{?}{b}$ act along a straight line in the same direction, then we can write $\left|\stackrel{?}{a}+\stackrel{?}{b}\right|$ = $\left|\stackrel{?}{a}\right|$ + $\left|\stackrel{?}{b}\right|\dots ..\left(v\right)$

Combining equations (iv) and (v), we get

$\left|\stackrel{?}{a}+\stackrel{?}{b}\right|$ $\le$ $\left|\stackrel{?}{a}\right|$ + $\left|\stackrel{?}{b}\right|$

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