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(D) The position vector of the centre of mass is given by $\vec{r}_{com} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1 + m_2}$.Substituting the given value

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$-2\hat{i}-\hat{j}+\hat{k}$

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(D) The position vector of the centre of mass is given by $\vec{r}_{com} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1 + m_2}$.Substituting the given values: $\vec{r}_{com} = \frac{1(\hat{i} + 2\hat{j} + \hat{k}) + 3(-3\hat{i} - 2\hat{j} + \hat{k})}{1 + 3}$.$\vec{r}_{com} = \frac{\hat{i} + 2\hat{j} + \hat{k} - 9\hat{i} - 6\hat{j} + 3\hat{k}}{4} = \frac{-8\hat{i} - 4\hat{j} + 4\hat{k}}{4} = -2\hat{i} - \hat{j} + \hat{k}$.The magnitude of the centre of mass position vector is $|\vec{r}_{com}| = \sqrt{(-2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$.Now,check the magnitude of the vector in option $D$: $|-2\hat{i} - \hat{j} + \hat{k}| = \sqrt{(-2)^2 + (-1)^2 + (1)^2} = \sqrt{6}$.Thus,the magnitudes are equal.

Two bodies of mass $1\,kg$ and $3\,kg$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of the position vector of the centre of mass of this system will be equal to the magnitude of which of the following vectors?

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