Question

When a right handed rectangular cartesian system $oxyz$ is rotated about the $z-$axis through an angle $\frac{\pi }{4}$ in the counter-clockwise direction it is found that a vector $\overrightarrow{a}$ has the components $2\sqrt{3},3\sqrt{2}$ and $4$.The component of $\overrightarrow{a}$ in the $oxyz$ coordinate system are

• A. $5,-1,4$
• B. $5,-1,4\sqrt{2}$
• C. $-1,-5,4\sqrt{2}$
• D. $-1,5,4$

Answer 1

The correct option is D $-1,5,4$

If $\hat{i},\hat{j},\hat{k}$ are unit vectors in the $oxyz$ system and $\hat{i},\hat{j},\hat{k}$ are the unit vectors in the system $o{x}^{\prime }{y}^{\prime }{z}^{\prime }$ obtained after rotation , then ${\hat{i}}_1=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j},{\hat{j}}_1=-\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}$ and ${\hat{k}}_1=\hat{k}$
$\overrightarrow{a}=2\sqrt{2}{\hat{i}}_1+3\sqrt{2}{\hat{j}}_1+4{\hat{k}}_1$
$=2\sqrt{2}(\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j})+3\sqrt{2}(\frac{-1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j})+4\hat{k}$
$=-\hat{i}+5\hat{j}+4\hat{k}$
So that the components in the $oxyz$ system are $-1,5,4$