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If \vec{a}\And \vec{b}are two unit vectors such that \vec{a}+2\vec{b}\And 5\vec{a}-4\vec{b}are perpendicular to each other, then angle between \vec{a}\And \vec{b} is,

Option: 1

{{45}^{\circ }}


Option: 2

{{60}^{\circ }}


Option: 3

{{\cos }^{-1}}(\frac{1}{3})


Option: 4

{{\cos }^{-1}}(\frac{2}{7})


Answers (1)

best_answer

\cos \theta =\frac{\vec{a}.\vec{b}}{\left| {\vec{a}} \right|\left| {\vec{b}} \right|} \\ (\vec{a}+2\vec{b}).(5\vec{a}-4\vec{b})=0 \\ 5{{\left| {\vec{a}} \right|}^{2}}-4(\vec{a}.\vec{b})+10(\vec{b}.\vec{a})-8{{\left| {\vec{b}} \right|}^{2}}=0 \\ \left| {\vec{a}} \right|=\left| {\vec{b}} \right|=1 \\ 5+6(\vec{a}.\vec{b})-8=0 \\ 6(\vec{a}.\vec{b})-3=0 \\ 6(\vec{a}.\vec{b})=3 \\ (\vec{a}.\vec{b})=\frac{1}{2} \\ \left| {\vec{a}} \right|\left| {\vec{b}} \right|\cos \theta =\frac{1}{2} \\ \cos \theta =\frac{1}{2} \\ \theta ={{60}^{\circ }} \\

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Pankaj

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