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Vector Algebra — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector Algebra5 Marks
Question
If $\vec a,\vec b,\vec c$ are unit vectors such that $\vec a + \vec b + \vec c = 0$ find the value of $\vec a.\vec b + \vec b.\vec c + \vec c.\vec a$.
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Answer

It is given that: If $\vec{a},\vec{b},\vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\vec{c}=0,$
then:
$(\vec a +\vec b+\vec c).(\vec a +\vec b+\vec c)=\vec 0 .\vec 0$
$\Rightarrow |\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a.\vec b+\vec b.\vec c+\vec c.\vec a)=0$
$\Rightarrow 1+1+1+2(\vec a.\vec b+\vec b.\vec c+\vec c.\vec a)=0$
$\Rightarrow (\vec a.\vec b+\vec b.\vec c+\vec c.\vec a)=-\frac{3}{2}$

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