Question

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ have magnitude $3$ each. $\overrightarrow{A}\times \overrightarrow{B}=-5\hat{k}+2\hat{i}$. Find the angle between $A$ and $B$.

• A. ${\cos }^{-1}\frac{\sqrt{29}}{9}$
• B. ${\tan }^{-1}\left(\frac{-5}{2}\right)$
• C. ${\sin }^{-1}\left(\frac{2}{5}\right)$
• D. ${\sin }^{-1}\left(\frac{\sqrt{29}}{9}\right)$

Answer 1

The correct option is D ${\sin }^{-1}\left(\frac{\sqrt{29}}{9}\right)$

The angle between vectors$\overrightarrow{A}$ and $\overrightarrow{B}$ is $sin\theta =\frac{|\overrightarrow{A}\times \overrightarrow{B}|}{\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|}$
But we are given $\overrightarrow{A}\times \overrightarrow{B}=5\hat{k}+2\hat{k}$
and $\left|\overrightarrow{A}\right|=\left|\overrightarrow{B}\right|=3$
Thus, $sin\theta =\frac{\sqrt{5^2+2^2}}{9}$
$\theta =si{n}^{-1}\left(\frac{\sqrt{29}}{9}\right)$