Three bodies of masses 5 kg, 4 kg and 2 kg have the position vectors as `2hati+3hatj-3hatk,hati+hatj+hatkand3hati-3hatj-4haktk`. Find the coordinates of centre of mass.

Answer

To find the coordinates of the center of mass for the three bodies with given masses and position vectors, we can follow these steps:

### Step 1: Identify the masses and position vectors
- Mass \( m_1 = 5 \, \text{kg} \) with position vector \( \vec{r_1} = 2\hat{i} + 3\hat{j} - 3\hat{k} \)
- Mass \( m_2 = 4 \, \text{kg} \) with position vector \( \vec{r_2} = \hat{i} + \hat{j} + \hat{k} \)
- Mass \( m_3 = 2 \, \text{kg} \) with position vector \( \vec{r_3} = 3\hat{i} - 3\hat{j} - 4\hat{k} \)

### Step 2: Calculate the total mass
The total mass \( M \) is given by:
\[ M = m_1 + m_2 + m_3 = 5 + 4 + 2 = 11 \, \text{kg} \]

### Step 3: Calculate the weighted position vector
The center of mass \( \vec{R} \) is calculated using the formula:
\[ \vec{R} = \frac{1}{M} \sum_{i=1}^{n} m_i \vec{r_i} \]
Substituting the values:
\[ \vec{R} = \frac{1}{11} \left( m_1 \vec{r_1} + m_2 \vec{r_2} + m_3 \vec{r_3} \right) \]

### Step 4: Substitute the values into the equation
Calculating each term:
\[ m_1 \vec{r_1} = 5(2\hat{i} + 3\hat{j} - 3\hat{k}) = 10\hat{i} + 15\hat{j} - 15\hat{k} \]
\[ m_2 \vec{r_2} = 4(\hat{i} + \hat{j} + \hat{k}) = 4\hat{i} + 4\hat{j} + 4\hat{k} \]
\[ m_3 \vec{r_3} = 2(3\hat{i} - 3\hat{j} - 4\hat{k}) = 6\hat{i} - 6\hat{j} - 8\hat{k} \]

### Step 5: Sum the components
Now, we sum the components:
\[ \vec{R} = \frac{1}{11} \left( (10 + 4 + 6)\hat{i} + (15 + 4 - 6)\hat{j} + (-15 + 4 - 8)\hat{k} \right) \]
Calculating each component:
- For \( \hat{i} \): \( 10 + 4 + 6 = 20 \)
- For \( \hat{j} \): \( 15 + 4 - 6 = 13 \)
- For \( \hat{k} \): \( -15 + 4 - 8 = -19 \)

### Step 6: Final calculation for center of mass
Now substituting back into the equation for \( \vec{R} \):
\[ \vec{R} = \frac{1}{11} (20\hat{i} + 13\hat{j} - 19\hat{k}) \]
Thus, the coordinates of the center of mass are:
\[ \vec{R} = \left( \frac{20}{11} \hat{i}, \frac{13}{11} \hat{j}, \frac{-19}{11} \hat{k} \right) \]

### Final Answer:
The coordinates of the center of mass are:
\[ \left( \frac{20}{11}, \frac{13}{11}, \frac{-19}{11} \right) \]