Question

If $\overrightarrow{ a }, \overrightarrow{ b }, \overrightarrow{ c }$ are non-coplanar unit vectors such that $\overrightarrow{ a } \times(\overrightarrow{ b } \times \overrightarrow{ c })=\frac{(\overrightarrow{ b }+\overrightarrow{ c })}{\sqrt{2}}$, then the angle between $\overrightarrow{ a }$ and $\overrightarrow{ b }$ is

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

Answer 1

Answer: (A)

Since, $ \overrightarrow{ a } \times(\overrightarrow{ b } \times \overrightarrow{ c })=\frac{\overrightarrow{ b }+\overrightarrow{ c }}{\sqrt{2}}$
$\Rightarrow (\vec{a} \cdot \overrightarrow{ c }) \overrightarrow{ b }-(\overrightarrow{ a } \cdot \overrightarrow{ b }) \overrightarrow{ c }=\frac{1}{\sqrt{2}} \overrightarrow{ b }+\frac{1}{\sqrt{2}} \overrightarrow{ c }$
On equating the coefficient of $\overrightarrow{ c }$, we get
$ \overrightarrow{ a } \cdot \overrightarrow{ b }=-\frac{1}{\sqrt{2}} \Rightarrow|\overrightarrow{ a }||\overrightarrow{ b }| \cos \theta=-\frac{1}{\sqrt{2}}$
$\therefore \cos \theta=-\frac{1}{\sqrt{2}} \Rightarrow \theta=\frac{3 \pi}{4}$