JEE Main & Advanced JEE Main Paper (Held On 22 April 2013)

  • question_answer
    If \[\hat{a},\hat{b}\] and \[\hat{c}\] are unit vectors satisfying \[\hat{a}-\sqrt{3}\] \[\hat{b}+\hat{c}=\overset{\to }{\mathop{0}}\,\] then the angle between the vectors \[\hat{a}\] and \[\hat{c}\] is :     JEE Main  Online Paper (Held On 22 April 2013)

    A)  \[\frac{\pi }{4}\]                                  

    B)  \[\frac{\pi }{3}\]

    C)  \[\frac{\pi }{6}\]                                  

    D)  \[\frac{\pi }{2}\]

    Correct Answer: B

    Solution :

     Let angle between \[\hat{a}\]and \[\hat{c}\] be \[\theta .\] Now, \[\hat{a}-\sqrt{3}\hat{b}+\hat{c}=\vec{0}\] \[\Rightarrow \]\[(\hat{a}+\hat{c})=\sqrt{3}\hat{b}\] \[\Rightarrow \]\[=(\hat{a}+\hat{c}).(\hat{a}+\hat{c})=3(\hat{b}.\hat{b})\] \[\Rightarrow \]\[\hat{a}.\hat{a}+\hat{a}.\hat{c}+\hat{c}.\hat{a}+\hat{c}.\hat{c}=3\times 1\] \[\Rightarrow \]\[1+2\cos \theta +1=3\] \[\Rightarrow \]\[\cos \theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}\]

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