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  4. The linear momentum of a particle overset arrow P=3 hati+4 hatj-2 hatk at a point with position vector is given by overset arrow r= hati+2 hatj- hatk . Then its angular momentum about the origin is perpendicular to,

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Q. The linear momentum of a particle $\overset{ \rightarrow }{P}=3\hat{i}+4\hat{j}-2\hat{k}$ at a point with position vector is given by $\overset{ \rightarrow }{r}=\hat{i}+2\hat{j}-\hat{k}$ . Then its angular momentum about the origin is perpendicular to,

NTA AbhyasNTA Abhyas 2020

Solution:

Here, $\overset{ \rightarrow }{r}=\hat{i}+2\hat{j}-\hat{k}, \, \overset{ \rightarrow }{p}=3\hat{i}+4\hat{j}-2\hat{k}$
$\overset{ \rightarrow }{L}=\overset{ \rightarrow }{r}\times \overset{ \rightarrow }{p}=\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -1 \\ 3 & 4 & -2 \end{vmatrix}$
$=\hat{i}\left(- 4 + 4\right)+\hat{j}\left(- 3 + 2\right)+\hat{k}\left(4 - 6\right)=0\hat{i}-1\hat{j}-2\hat{k}$
$\overset{ \rightarrow }{L}$ has components along $-y$ axis but it has no component along in the $x-$ axis. The angular momentum $\overset{ \rightarrow }{L}$ is in $yz$ -plane. $i.e.$ perpendicular to $x-$ axis.