Question

A vector of length $\sqrt{7}$ which is perpendicular to $2\overline{j}-\overline{k}$ and $-\overline{i}+2\overline{j}-3\overline{k}$ and makes obtuse angle with y-axis is

• A. $\frac{1}{\sqrt{5}}(4\overline{i}-\overline{j}+\sqrt{18}\overline{k})$
• B. $\frac{1}{\sqrt{3}}(4\overline{i}-\overline{j}-2\overline{k})$
• C. $\frac{1}{\sqrt{3}}(-4\overline{i}+\overline{j}+2\overline{k})$
• D. $\frac{1}{\sqrt{3}}(-4\overline{i}-\overline{j}+2\overline{k})$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}(-4\overline{i}+\overline{j}+2\overline{k})$

A vector perpendicular to $A=2\overrightarrow{j}-\overrightarrow{k}$ and $-\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}$ is $\overrightarrow{A}\times \overrightarrow{B}$

$\overrightarrow{A}\times \overrightarrow{B}=-4\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}$

The unit vector perpendicular to $\overrightarrow{A}$ and $\overrightarrow{B}$ is

$\frac{-4\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}}{\sqrt{21}}$

The vector of length $\sqrt{7}$ perpendicular to $\overrightarrow{A}$ and $\overrightarrow{B}$ is,

$\left(\frac{-4\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}}{\sqrt{21}}\right)\times \sqrt{7}=\frac{-4\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}}{\sqrt{3}}$

Option c gives the correct answer