Question

Let $\stackrel{\_}{v}$ be a unit vector which follows the equation,$\stackrel{\_}{v}\times \stackrel{\_}{b}=\stackrel{\_}{c}$. also $\left|\stackrel{\_}{b}\right|=2$ and $\left|\stackrel{\_}{c}\right|=\sqrt{3}$ then

• A. $\stackrel{\_}{v=-}\stackrel{\_}{b}+\stackrel{\_}{b}\times \stackrel{\_}{c}$
• B. $\stackrel{\_}{v}=\frac{3}{4}(\stackrel{\_}{b}+2\stackrel{\_}{b}\times \stackrel{\_}{c})$
• C. $\stackrel{\_}{v}=\frac{1}{4}(\stackrel{\_}{b}+\stackrel{\_}{b}\times \stackrel{\_}{c})$
• D. none of these

Answer 1

Answer: (C)

$\stackrel{\_}{v}=\frac{1}{4}(\stackrel{\_}{b}+\stackrel{\_}{b}\times \stackrel{\_}{c})$

$\overrightarrow{v}\times \overrightarrow{b}=\overrightarrow{c}$

$\text{and}\overrightarrow{v}\text{is a unit vector.}$

$|\overrightarrow{v}|=1$

$|\overrightarrow{b}|=2$

$|\overrightarrow{c}|=\sqrt{3}$

$|\overrightarrow{v}\times \overrightarrow{b}|=|\overrightarrow{c}|$

$|\overrightarrow{v}||\overrightarrow{b}|\sin \theta =|\overrightarrow{c}\mid$

$1\cdot 2\sin \theta =3$

$\sin \theta =\frac{\sqrt{3}}{2},\cos \theta =\frac{1}{2}$

$\overrightarrow{b}\times (\overrightarrow{v}\times \overrightarrow{b})=\overrightarrow{c}\times \overrightarrow{b}$

$(\overrightarrow{b}\cdot \overrightarrow{v})\overrightarrow{b}-(\overrightarrow{b}\cdot \overrightarrow{b})\overrightarrow{v}=\overrightarrow{c}\times \overrightarrow{b}$

$(\begin{array}{ccc}|b||v|\cos \theta )\overrightarrow{b}& -(1\overrightarrow{b}\mid {)}^2\nabla & =\overrightarrow{c}\times \overrightarrow{b}\\ 1\cdot 2\cdot \frac{1}{2}\overrightarrow{b}-\left(4\right)\overrightarrow{v}& =\overrightarrow{c}\times \overrightarrow{b}& \end{array}$

$\overrightarrow{b}-\overrightarrow{c}\times \overrightarrow{b}=4\overrightarrow{v}$

$\frac{1}{4}(\overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c})$

option $C:\overrightarrow{v}=\frac{1}{4}(\overrightarrow{b}+\overline{b}\times \overrightarrow{c})$