Question
Let $$\overrightarrow a ,\,\overrightarrow b ,\,\overrightarrow c $$ be three unit vectors of which $$\overrightarrow b $$ and $$\overrightarrow c $$
are nonparallel. Let the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ be $$\alpha $$ and that between $$\overrightarrow a $$ and $$\overrightarrow c $$ be $$\beta .$$ If $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = \frac{1}{2}\overrightarrow b $$ then :
A.
$$\alpha = \frac{\pi }{3},\,\,\beta = \frac{\pi }{2}$$
B.
$$\alpha = \frac{\pi }{2},\,\,\beta = \frac{\pi }{3}$$
C.
$$\alpha = \frac{\pi }{6},\,\,\beta = \frac{\pi }{3}$$
D.
none of these
Answer :
$$\alpha = \frac{\pi }{2},\,\,\beta = \frac{\pi }{3}$$
Solution :
$$\eqalign{
& \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = \frac{1}{2}\overrightarrow b \,\,\,\,\,\, \Rightarrow \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow b - \left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow c = \frac{1}{2}\overrightarrow b \cr
& \therefore \,\overrightarrow a .\overrightarrow c = \frac{1}{2},\,\,\,\,\overrightarrow a .\overrightarrow b = 0 \cr
& \therefore \,\left| {\overrightarrow a } \right|\left| {\overrightarrow c } \right|\cos \,\beta = \frac{1}{2},\,\,\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\cos \,\alpha = 0 \cr
& {\text{or }}\cos \,\beta = \frac{1}{2},\,\,\cos \,\alpha = 0\,\,\, \Rightarrow \,\beta = \frac{\pi }{3},\,\alpha = \frac{\pi }{2} \cr} $$