Establish the following vector inequalities geometrically or otherwise:

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Establish the following vector inequalities geometrically or otherwise:

$\left| \overrightarrow{a}+\overrightarrow{b} \right|\le \left| \overrightarrow{a} \right|+\left| {\vec{b}} \right|$

Answer –

Let two vectors $\vec{a}$ and $\vec{b}$ represent two adjacent sides of a parallelogram PQRS, as show in the diagram below :

Here,

$\left| \overrightarrow{SR} \right|=|\vec{a}|$ and $\left| \overrightarrow{QR} \right|=\left| {\vec{b}} \right|$

Also, $\left| \overrightarrow{PR} \right|=\left| \overrightarrow{a}+\overrightarrow{b} \right|$

We know that for a triangle, each side is always smaller than the sum of other two sides. So, we can write

PR < QR + SR

$\left| \overrightarrow{a}+\overrightarrow{b} \right|<\left| \overrightarrow{a} \right|+\left| {\vec{b}} \right|\to (1)$

Now, if both the said vectors act in a straight line then we can write,

$\left| \overrightarrow{a}+\overrightarrow{b} \right|=\left| \overrightarrow{a} \right|+\left| {\vec{b}} \right|\to (2)$

Upon combining (1) and (2), we get the required result as,

$\left| \overrightarrow{a}+\overrightarrow{b} \right|\le \left| \overrightarrow{a} \right|+\left| {\vec{b}} \right|$

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