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એક મચ્છર $\overrightarrow{ v }=0.5 t ^{2} \hat{ i }+3 t \hat{ j }+9 \hat{ k }\, m / s$ ના વેગથી અને સમાન પ્રવેગથી ગતિ કરે છે. $2 \,s$ ના અંતે મચ્છરની દિશા કઈ હશે ?
$x-$ અક્ષથી $\tan ^{-1}\left(\frac{2}{3}\right)$
$y$ -અક્ષથી $\tan ^{-1}\left(\frac{2}{3}\right)$
$y$ -અક્ષથી $\tan ^{-1}\left(\frac{5}{2}\right)$
$x$ -અક્ષથી $\tan ^{-1}\left(\frac{5}{2}\right)$
Solution
Given :
$\overrightarrow{ v }=0.5 t ^{2} \hat{ i }+3 t \hat{ j }+9 \hat{ k }$
$\overrightarrow{ v }_{ at\,t =2}=2 \hat{ i }+6 \hat{ j }+9 \hat{ k }$
$\therefore$ Angle made by direction of motion of mosquito will be,
$\cos ^{-1} \frac{2}{11}($ from $x$ -axis $)=\tan ^{-1} \frac{\sqrt{117}}{2}$
$\cos ^{-1} \frac{6}{11}($ from $y$ -axis $)=\tan ^{-1} \frac{\sqrt{85}}{6}$
$\cos ^{-1} \frac{9}{11}($ from $z$ -axis $)=\tan ^{-1} \frac{\sqrt{40}}{9}$
None of the option is matching.
Hence this question should be bonus.
