Question

If $ \overrightarrow{a} $ and $ \overrightarrow{b} $ are unit vectors such that $ [\overrightarrow{a}\,\overrightarrow{b}\,\overrightarrow{a}\times \overrightarrow{b}]=\frac{1}{4}, $ then angle between $ \overrightarrow{a} $ and $ \overrightarrow{b} $ is:

• A. $ \frac{\pi }{3} $
• B. $ \frac{\pi }{4} $
• C. $ \frac{\pi }{6} $
• D. $ \frac{\pi }{2} $
• E. $ \frac{\pi }{12} $

Answer 1

Answer: (C)

$ |\overrightarrow{a}|=1,|\overrightarrow{b}|=1,[\overrightarrow{a},\overrightarrow{b},\overrightarrow{a}\times \overrightarrow{b}]=\frac{1}{4} $ $ \Rightarrow $ $ (\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{b})=\frac{1}{4} $ $ \Rightarrow $ $ {{(|\overrightarrow{a}||\overrightarrow{b}|\sin \theta )}^{2}}=\frac{1}{4}\Rightarrow {{\sin }^{2}}\theta =\frac{1}{4} $ $ \Rightarrow $ $ \sin \theta =\frac{1}{2} $ $ \Rightarrow $ $ \theta =\frac{\pi }{6} $