Question

The angle between $ \overrightarrow{a} $ and $ \overrightarrow{b} $ is $ \frac{5\pi }{6} $ and the projection of $ \overrightarrow{a} $ in the direction of $ \overrightarrow{b} $ is $ \frac{-6}{\sqrt{3}}, $ then $ |\overrightarrow{a}| $ is equal to:

• A. 6
• B. $ \frac{\sqrt{3}}{2} $
• C. 12
• D. 4
• E. 16

Answer 1

Answer: (D)

$ \because $ $ \overrightarrow{a}.\overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \frac{5\pi }{6} $
$ =-\frac{|\overrightarrow{a}||\overrightarrow{b}|\sqrt{3}}{2} $
Since, the projection of $ \overrightarrow{a} $ in the direction of $ \overrightarrow{b} $ $ \therefore $ $ -\frac{6}{\sqrt{3}}=-\frac{|\overrightarrow{a}||\overrightarrow{b}|\sqrt{3}}{2|\overrightarrow{b}|} $
$ \Rightarrow $ $ |\overrightarrow{a}|=\frac{6\times 2}{3}=4 $