If a , b , c are unit vectors such that a , b , c =0, then write … - Vidyadip

Question

If $\vec{\text{a}}, \text{ }\vec{\text{b}},\text{ } \vec{\text{c}}$ are unit vectors such that $\vec{\text{a}}, \text{ }\vec{\text{b}}, \text{ }\vec{\text{c}}\text{ }=\vec{0},$ then write the value of $\vec{\text{a}} \text{ . }\vec{\text{b}} + \vec{\text{b}} \text{ . } \text{ }\vec{\text{c}} +\vec{\text{c}} . \vec{\text{a}}\text{ }.$

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Answer

$(\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) . (\vec{\text{a}} + \vec{\text{b}} + \vec{\text{c}}) = 0$
$\Rightarrow |\vec{\text{a}}|^{2} + |\vec{\text{b}}|^{2} + |\vec{\text{c}}|^{2} + 2 (\vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}}) = 0$
$\Rightarrow \vec{\text{a}}.\vec{\text{b}} + \vec{\text{b}}.\vec{\text{c}} + \vec{\text{c}}.\vec{\text{a}} = -\frac{3}{2}$

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