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in Vector algebra by (20 points)
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Suppose in a 'Methane molecule' \( \left( CH _{4}\right) \) each hydrogen atom is at the corner of a tetrahedron with the carbon atom at the centre. With respect to carbon atom as origin let \( \vec{A}=\left(\log _{3}|\lambda|\right) \hat{i}+\hat{j}-\left(\log _{3} \lambda^{2}\right) \hat{k} \) and \( \vec{B}=2 \hat{i}+\hat{j}-\left(\log _{3}|\lambda|\right) \hat{k} \) where \( \lambda \in[-3,-1] \) \( \cup[1,9] \) represents \( C - Ha \& C - H _{\beta} \) bonds respectively (as shown in the figure and \( Ha , H _{\beta}, H _{\gamma}, H _{\delta} \) denotes four hydrogen atom of methane. Also the bond length will vary according to the value of \( \lambda \) ). If \( L= \) maximum distance between \( H _{ a } \& H _{\beta}, l= \) minimum distance between \( H _{ a } \& H _{\beta} \), then \( ( L + l ) \) is equal to-

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