Question

Two vectors $\overrightarrow{a}=\overrightarrow{i}+\frac{\overrightarrow{j}}{\sqrt{3}}$ and $\overrightarrow{b}=\frac{\overrightarrow{i}}{\sqrt{3}}+\overrightarrow{j}$ are

• A. Perpendicular to each other
• B. Parallel to each other
• C. Inclined to each other at an angle $\frac{\pi }{3}$
• D. Inclined to each other at an angle $\frac{\pi }{6}$

Answer 1

Answer: (D)

Inclined to each other at an angle $\frac{\pi }{6}$

$\overrightarrow{a}.\overrightarrow{b}=a.b.\cos \alpha$

$\Rightarrow (i+\frac{j}{\sqrt{3}}).(\frac{i}{\sqrt{3}}+j)=\frac{2}{\sqrt{3}}\frac{2}{\sqrt{3}}.\cos \alpha$

$\Rightarrow \frac{2}{\sqrt{3}}=\frac{2}{\sqrt{3}}\frac{2}{\sqrt{3}}.\cos \alpha$

$\Rightarrow \cos \alpha =\frac{\sqrt{3}}{2}$

$\Rightarrow \alpha =\frac{\pi }{6}$