question_answer

If \[\overline{a},\,\,\bar{b},\,\,\bar{c}\,\,and\,\,\bar{d}\] are unit vector such that \[(\vec{a}\,\times \vec{b}).(\vec{c}\,\times \vec{d})=1\] and \[\vec{a}.\vec{c}=\frac{1}{2}\], then

A) \[\vec{a},\,\,\vec{b},\,\,\vec{c}\] are non-coplanar

B) \[\vec{b},\,\,\vec{c},\,\,\vec{d}\] are non-coplanar

C) \[\vec{b},\,\,\vec{d}\] are non-parallel

D) \[\vec{a},\,\,\vec{d}\] are parallel and \[\vec{b},\,\,\vec{c}\] are parallel

Correct Answer: C

Solution :

\[\vec{a}\times \vec{b}= \left| {\vec{a}} \right|\left| {\vec{b}} \right| \sin \,\,\alpha \,\vec{n}=\sin \,\alpha \,{{\vec{n}}_{1}},\alpha \in \,\left[ 0,\,\,\pi  \right]\] \[\vec{c}\,\times \vec{d}=sin\,\beta \,\,{{\vec{n}}_{2}},\,\,\beta \in [0,\,\,\pi ]\] Now \[\left( \vec{a}\,\times \vec{b} \right)\,.\left( \vec{c}\,\times \vec{d} \right)=1\] \[\Rightarrow \,\,\,sin\,\alpha .sin\,\beta \,({{\hat{n}}_{1}},.{{\hat{n}}_{2}})=1,\] \[\Rightarrow \,\,\,sin\,\alpha \,.\,\,sin\,\beta \,cos\,\theta =1\] Where \[\theta \] is the angle between n, and m \[\Rightarrow \,\,\,\alpha =\pi /2,\,\,\beta =\pi /2 \,and\,\,\theta =0\] Now, \[\vec{a}\,.\,\vec{c}=\frac{1}{2}\] \[\Rightarrow \,\,\,cos\,\gamma =1/2\,\,\,\Rightarrow \,\,\,\gamma =\pi /3\] As \[\bar{a}\times \bar{b}\parallel \bar{c}\times \bar{d},\,\,\bar{a},\bar{b},\bar{c},\bar{d}\] are coplanar. There are two possibilities as shown in figure. Thus \[\vec{b}\,\,and\,\,\vec{d}\] are non-parallel