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Addition of Vectors

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Question

If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors then $|\vec{a} + \vec{b} + \vec{c}|$ = _______________.

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Solution

Given:
$\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular vectors
then, $\vec{a} \cdot \vec{b} = 0, \vec{a} \cdot \vec{c} = 0 and \vec{c} \cdot \vec{b} = 0$

${|\vec{a} + \vec{b} + \vec{c}|}^{2} = \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} + \vec{c} \cdot \vec{c} + 2 \vec{a} \cdot \vec{b} + 2 \vec{b} \cdot \vec{c} + 2 \vec{c} \cdot \vec{a} = {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 (\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} + 2 (0 + 0 + 0) (∵ \vec{a} \cdot \vec{b} = 0, \vec{a} \cdot \vec{c} = 0 and \vec{c} \cdot \vec{b} = 0) = {|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2} \Rightarrow |\vec{a} + \vec{b} + \vec{c}| = \sqrt{{|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2}}$

Hence, $|\vec{a} + \vec{b} + \vec{c}| = \sqrt{{|\vec{a}|}^{2} + {|\vec{b}|}^{2} + {|\vec{c}|}^{2}}$ .

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