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The moment about the point $\hat{i}+2 \hat{j}+3 \hat{k}$ of a force represented by $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ acting through the point $2 \mathrm{i}+3 \mathrm{j}+\mathrm{k}$ is
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Verified Answer
The correct answer is:
$3 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}$
Here, $r=(2 \hat{i}+3 \hat{j}+\hat{k})-(i+2 \hat{j}+3 \hat{k})$
$\Rightarrow \mathrm{r}=\mathrm{i}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \text { and } \mathrm{F}=\mathrm{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$
Then, the required moment is given by
$r \times F=(i+\hat{j}-2 \hat{k}) \times(i+\hat{j}+\hat{k})$
$=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -2 \\ 1 & 1 & 1\end{array}\right|=3 \hat{i}-3 \hat{j}$
$\therefore$ Moment about given point $=3 \hat{i}-3 \hat{j}$
$\Rightarrow \mathrm{r}=\mathrm{i}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}} \text { and } \mathrm{F}=\mathrm{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$
Then, the required moment is given by
$r \times F=(i+\hat{j}-2 \hat{k}) \times(i+\hat{j}+\hat{k})$
$=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & -2 \\ 1 & 1 & 1\end{array}\right|=3 \hat{i}-3 \hat{j}$
$\therefore$ Moment about given point $=3 \hat{i}-3 \hat{j}$
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