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 centre of mass of this system has a position vector.

Options

A. `-2 hat i+ 2 hat k`
B. `- 2 hat i- hat j + hat k`
C. `2 hat i- hat j - 2 hat k`
D. `-1 hat i+ hat j + hat k`

Explanation

To find the center of mass of the two bodies, we can use the formula for the center of mass (R) of a system of particles:

\[ \mathbf{R} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2}}{m_1 + m_2} \]

where:
- \( m_1 \) and \( m_2 \) are the masses of the two bodies,
- \( \mathbf{r_1} \) and \( \mathbf{r_2} \) are the position vectors of the two bodies.

### Step 1: Identify the given values
- Mass of the first body, \( m_1 = 1 \, \text{kg} \)
- Position vector of the first body, \( \mathbf{r_1} = \hat{i} + 2\hat{j} + \hat{k} \)
- Mass of the second body, \( m_2 = 3 \, \text{kg} \)
- Position vector of the second body, \( \mathbf{r_2} = -3\hat{i} - 2\hat{j} + \hat{k} \)

### Step 2: Substitute the values into the formula
\[ \mathbf{R} = \frac{1 \cdot (\hat{i} + 2\hat{j} + \hat{k}) + 3 \cdot (-3\hat{i} - 2\hat{j} + \hat{k})}{1 + 3} \]

### Step 3: Calculate the numerator
Calculate the contributions from each mass:
- Contribution from the first body:
\[ 1 \cdot (\hat{i} + 2\hat{j} + \hat{k}) = \hat{i} + 2\hat{j} + \hat{k} \]

- Contribution from the second body:
\[ 3 \cdot (-3\hat{i} - 2\hat{j} + \hat{k}) = -9\hat{i} - 6\hat{j} + 3\hat{k} \]

Now, combine these contributions:
\[ \hat{i} + 2\hat{j} + \hat{k} - 9\hat{i} - 6\hat{j} + 3\hat{k} = (-8\hat{i} - 4\hat{j} + 4\hat{k}) \]

### Step 4: Calculate the total mass
The total mass of the system is:
\[ m_1 + m_2 = 1 + 3 = 4 \, \text{kg} \]

### Step 5: Divide the combined vector by the total mass
\[ \mathbf{R} = \frac{-8\hat{i} - 4\hat{j} + 4\hat{k}}{4} \]
\[ \mathbf{R} = -2\hat{i} - \hat{j} + \hat{k} \]

### Final Answer
The position vector of the center of mass is:
\[ \mathbf{R} = -2\hat{i} - \hat{j} + \hat{k} \]

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