Question

If momenta of two particles of a system are given by: ${\overrightarrow{p}}_1=2\hat{i}-\hat{j}+3\hat{k}$ and ${\overrightarrow{p}}_2=-\hat{i}+2\hat{j}+3\hat{k}$, then the angle made by the direction of motion of the system with $x-axis$ is :

• A. ${\cos }^{-1}\left(1\right)$
• B. ${\cos }^{-1}\sqrt{\frac{36}{38}}$
• C. ${45}^{\circ }$
• D. ${\cos }^{-1}\sqrt{\frac{1}{38}}$

Answer 1

The correct option is D ${\cos }^{-1}\sqrt{\frac{1}{38}}$

Correct option is $\left(D\right)$

$\overrightarrow{P_1}=2\hat{i}-\hat{j}+3\hat{k}$ and ${\overrightarrow{P}}_2=-\hat{i}+2\hat{j}+3\hat{k}$

System of

$\frac{1}{{\overrightarrow{P}}_{\text{system}}}={\overrightarrow{P}}_1+{\overrightarrow{P}}_2$

$=\hat{i}+\hat{j}+6\hat{k}$

$\overrightarrow{{\hat{P}}_S\cdot \hat{i}}=\left|{\overrightarrow{P}}_s\right||\hat{i}|\cos \theta$

$\begin{array}{cc}1& =\sqrt{38}\cos \theta \\ {\cos }^{-1}\left(\sqrt{\frac{1}{38}}\right)& =\theta \end{array}$