Question

The vector $\overline{OP}=\hat{i}+2\hat{j}+2\hat{k}$ turns through a right angle passing through the positive x-axis on the way. Find the vector in its new position.

• A. $\frac{4}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k}$
• B. $-\frac{4}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k}$
• C. $-\frac{4}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}$
• D. $\frac{4}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k}$

Answer 1

The correct option is A $\frac{4}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k}$

$\overrightarrow{OP}=\hat{i}+2\hat{j}+2\hat{k}$
After rotation of $\overrightarrow{OP}$, let new vector is $\overrightarrow{O{P}^{\prime }}$
Now $\overrightarrow{OP},\hat{i},\overrightarrow{O{P}^{\prime }}$ will be coplanar .
So $\overrightarrow{O{P}^{\prime }}=\left|\overrightarrow{OP}\right|\frac{(\overrightarrow{OP}\times \hat{i})\times \overrightarrow{OP}}{|(\overrightarrow{OP}\times \hat{i})\times \overrightarrow{OP}|}[\because \left|\overrightarrow{O{P}^{\prime }}\right|=\left|\overrightarrow{OP}\right|]$
But $(\overrightarrow{OP}\times \hat{i})\times \overrightarrow{OP}=8\hat{i}-2\hat{j}-2\hat{k}$
$\Rightarrow \overrightarrow{O{P}^{\prime }}=\frac{3(8\hat{i}-2\hat{j}-2\hat{k})}{2\times 3\sqrt{2}}$ or $\overrightarrow{O{P}^{\prime }}=\frac{4}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}-\frac{1}{\sqrt{2}}\hat{k}$