Question

The angle which the vector $\overrightarrow{A}=2\hat{i}+3\hat{j}$ makes with the y-axis, where $\hat{i}$ and $\hat{j}$ are unit vectors along x- and y-axes, respectively, is

• A. ${\cos }^{-1}\left(\frac{3}{5}\right)$
• B. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• C. ${\tan }^{-1}\left(\frac{2}{3}\right)$
• D. ${\sin }^{-1}\left(\frac{2}{3}\right)$

Answer 1

The correct option is C ${\tan }^{-1}\left(\frac{2}{3}\right)$

$\overrightarrow{A}=2\hat{i}+3\hat{j}$

$\left|\overrightarrow{A}\right|=\sqrt{4+9}=\sqrt{13}$

$\sin \theta =\frac{x}{\left|\overrightarrow{A}\right|}=\frac{2}{\sqrt{13}}$

$\cos \theta =\frac{y}{\left|\overrightarrow{A}\right|}=\frac{3}{\sqrt{13}}$

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{2}{\sqrt{13}}\times \frac{\sqrt{13}}{3}=\frac{2}{3}$

$\theta ={\tan }^{-1}\left(\frac{2}{3}\right)$