Question

The resultant moment of three forces $\hat{i}+2\hat{j}-3\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $-\hat{i}-\hat{j}+\hat{k}$ acting on a particle at a point $P$ $(0,1,2)$ about the point $A(1,-2,0)$ is

• A. $6\sqrt{2}$
• B. $\sqrt{140}$
• C. $\sqrt{21}$
• D. None

Answer 1

Answer: (B)

If $\vec{F}$ be the resultant of the three given forces then
$\vec{F}=(\hat{i}+2\hat{j}-3\hat{k})+(2\hat{i}+3\hat{j}+4\hat{k})+(-\hat{i}-\hat{j}+\hat{k})=2\hat{i}+4\hat{j}+2\hat{k}$
If $O$ be the origin, then $\overrightarrow{OP}=p.v$. of $P=\hat{j}+2\hat{k}$
$\overrightarrow{OA}=p.v.$ of $A=\hat{i}-2\hat{j}$
$\therefore \overrightarrow{AP}=\overrightarrow{OP}-\overrightarrow{OA}=(\hat{j}+2\hat{k})-(\hat{i}-2\hat{j})=-\hat{i}+3\hat{j}+2\hat{k}$
$\therefore$ Vector moment of the given forces about
$A=$ vector moment of $\vec{F}$ about $A=\overrightarrow{AP}\times \overrightarrow{F}$
$=(-\hat{i}+3\hat{j}+2\hat{k})\times (2\hat{i}+4\hat{j}+2\hat{k})=-2\hat{i}+6\hat{j}-10\hat{k}$
The magnitude of the moment $=\sqrt{4+36+100}=\sqrt{140}$