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Let \vec{a}, \vec{b}and \vec{c}be three non zero vectors such that \vec{b} \cdot \vec{c}=0 and \vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}.If \vec{d}be a vector such that \vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b},then (\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})is equal to

 

Option: 1

-\frac{1}{4}


Option: 2

\frac{1}{4}


Option: 3

\frac{3}{4}


Option: 4

\frac{1}{2}


Answers (1)

best_answer

(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{b}}-(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}) \overline{\mathrm{c}}=\frac{\overline{\mathrm{b}}}{2}-\frac{\overline{\mathrm{c}}}{2}
\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=\frac{1}{2}, \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=\frac{1}{2}
\overline{\mathrm{b}} \cdot \overline{\mathrm{d}}=\frac{1}{2}

(\overline{\mathrm{a}} \times \overline{\mathrm{b}}) \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{d}})=\overline{\mathrm{a}} \cdot[\overline{\mathrm{b}} \times(\overline{\mathrm{c}} \times \overline{\mathrm{d}})]
=\overline{\mathrm{a}} \cdot[(\overline{\mathrm{b}} \cdot \overline{\mathrm{d}}) \overline{\mathrm{c}}-(\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}) \overline{\mathrm{d}}]
=\overline{\mathrm{a}} \cdot[\overline{\mathrm{c}} / 2]
=\frac{1}{2}(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}})
=\frac{1}{4}
 

Posted by

Irshad Anwar

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