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)+2hat(k)`
B. `-2hat(i)-hat(j)+hat(k)`
C. `2hat(i)-hat(j)-2hat(k)`
D. `-hat(i)+hat(j)+hat(k)`

Answer

To find the position vector of the center of mass of the two bodies, we can use the formula for the center of mass (CM) given by:

\[ \vec{R}_{CM} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2} \]

where:
- \(m_1\) and \(m_2\) are the masses of the two bodies,
- \(\vec{r}_1\) and \(\vec{r}_2\) are the position vectors of the two bodies.

### Step 1: Identify the masses and position vectors
Given:
- Mass \(m_1 = 1 \, \text{kg}\)
- Position vector \(\vec{r}_1 = \hat{i} + 2\hat{j} + \hat{k}\)
- Mass \(m_2 = 3 \, \text{kg}\)
- Position vector \(\vec{r}_2 = -3\hat{i} - 2\hat{j} + \hat{k}\)

### Step 2: Substitute the values into the center of mass formula
Now, we substitute the values into the formula:

\[ \vec{R}_{CM} = \frac{1 \cdot (\hat{i} + 2\hat{j} + \hat{k}) + 3 \cdot (-3\hat{i} - 2\hat{j} + \hat{k})}{1 + 3} \]

### Step 3: Calculate the numerator
Calculating the numerator:

1. For \(m_1 \vec{r}_1\):
\[ 1 \cdot (\hat{i} + 2\hat{j} + \hat{k}) = \hat{i} + 2\hat{j} + \hat{k} \]

2. For \(m_2 \vec{r}_2\):
\[ 3 \cdot (-3\hat{i} - 2\hat{j} + \hat{k}) = -9\hat{i} - 6\hat{j} + 3\hat{k} \]

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

### Step 4: Calculate the denominator
The denominator is:
\[ m_1 + m_2 = 1 + 3 = 4 \]

### Step 5: Divide the numerator by the denominator
Now, we can find the center of mass position vector:
\[ \vec{R}_{CM} = \frac{-8\hat{i} - 4\hat{j} + 4\hat{k}}{4} \]
\[ = -2\hat{i} - \hat{j} + \hat{k} \]

### Final Answer
Thus, the position vector of the center of mass is:
\[ \vec{R}_{CM} = -2\hat{i} - \hat{j} + \hat{k} \]

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