# Is | vector a − vector b | greater than or less than | vector a | + | vector b |?

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Is | $\vec a-\vec b$ | greater than or less than | $\vec a$ | + | $\vec b$ |?

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${|\vec a-\vec b|}^2-{(|\vec a|+|\vec b|)}^2$

$|\vec a|^2+|\vec b|^2-2\,|\vec a|\,|\vec b|\,cos\, \theta-|\vec a|^2-|\vec b|^2-2|\vec a||\vec b|$

⇒-4| a | | b | $cos^2\frac{\theta}{2}$

= a negative quantity

Hence,  $|\vec a- \vec b|<|\vec a|+|\vec b|$