Two particles of mass `1 kg and 3 kg` have position vectors `2 hat i+ 3 hat j + 4 hat k and -2 hat i+ 3 hat j - 4 hat k` respectively. The centre of mass has a position vector.

Options

A. `hat i+ 3 hat j - 2 hat k`
B. ` - hat i- 3 hat j - 2 hat k`
C. `- hat i+ 3 hat j + 2 hat k`
D. ` - hat i+ 3 hat j - 2 hat k`

Explanation

To find the position vector of the center of mass of the two particles, we can follow these steps:

### Step 1: Identify the masses and position vectors
- The mass of the first particle \( m_1 = 1 \, \text{kg} \) with position vector \( \vec{r_1} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \).
- The mass of the second particle \( m_2 = 3 \, \text{kg} \) with position vector \( \vec{r_2} = -2 \hat{i} + 3 \hat{j} - 4 \hat{k} \).

### Step 2: Use the formula for the center of mass
The position vector \( \vec{R} \) of the center of mass is given by the formula:
\[ \vec{R} = \frac{m_1 \vec{r_1} + m_2 \vec{r_2}}{m_1 + m_2} \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ \vec{R} = \frac{1 \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + 3 \cdot (-2 \hat{i} + 3 \hat{j} - 4 \hat{k})}{1 + 3} \]

### Step 4: Calculate the numerator
Calculating the first part:
\[ 1 \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
Calculating the second part:
\[ 3 \cdot (-2 \hat{i} + 3 \hat{j} - 4 \hat{k}) = -6 \hat{i} + 9 \hat{j} - 12 \hat{k} \]
Now combine these:
\[ (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + (-6 \hat{i} + 9 \hat{j} - 12 \hat{k}) = (2 - 6) \hat{i} + (3 + 9) \hat{j} + (4 - 12) \hat{k} \]
This simplifies to:
\[ -4 \hat{i} + 12 \hat{j} - 8 \hat{k} \]

### Step 5: Divide by the total mass
Now, the total mass \( m_1 + m_2 = 1 + 3 = 4 \):
\[ \vec{R} = \frac{-4 \hat{i} + 12 \hat{j} - 8 \hat{k}}{4} \]
This simplifies to:
\[ \vec{R} = -1 \hat{i} + 3 \hat{j} - 2 \hat{k} \]

### Final Answer
Thus, the position vector of the center of mass is:
\[ \vec{R} = -1 \hat{i} + 3 \hat{j} - 2 \hat{k} \]

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