The position vector of three particles of masses `m_1=1kg`.
`m_2=2kg` and `m_3=3kg` are `r_1=(hati+4hatj+hatk)m`, `r_2=(hati+hatj+hatk)m` and `r_3=(2hati-hatj-2hatk)m` respectively. Find the position vector of their centre of mass.

Explanation

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

### Step 1: Identify the given values
We have three particles with the following masses and position vectors:
- Mass \( m_1 = 1 \, \text{kg} \) and position vector \( \mathbf{r_1} = \hat{i} + 4\hat{j} + \hat{k} \, \text{m} \)
- Mass \( m_2 = 2 \, \text{kg} \) and position vector \( \mathbf{r_2} = \hat{i} + \hat{j} + \hat{k} \, \text{m} \)
- Mass \( m_3 = 3 \, \text{kg} \) and position vector \( \mathbf{r_3} = 2\hat{i} - \hat{j} - 2\hat{k} \, \text{m} \)

### Step 2: Write the formula for the center of mass
The position vector of the center of mass \( \mathbf{r_{cm}} \) is given by the formula:
\[ \mathbf{r_{cm}} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2} + m_3 \mathbf{r_3}}{m_1 + m_2 + m_3} \]

### Step 3: Calculate the total mass
First, we calculate the total mass:
\[ m_1 + m_2 + m_3 = 1 + 2 + 3 = 6 \, \text{kg} \]

### Step 4: Calculate the numerator
Now we calculate the numerator \( m_1 \mathbf{r_1} + m_2 \mathbf{r_2} + m_3 \mathbf{r_3} \):
- For \( m_1 \mathbf{r_1} \):
\[ m_1 \mathbf{r_1} = 1 \cdot (\hat{i} + 4\hat{j} + \hat{k}) = \hat{i} + 4\hat{j} + \hat{k} \]

- For \( m_2 \mathbf{r_2} \):
\[ m_2 \mathbf{r_2} = 2 \cdot (\hat{i} + \hat{j} + \hat{k}) = 2\hat{i} + 2\hat{j} + 2\hat{k} \]

- For \( m_3 \mathbf{r_3} \):
\[ m_3 \mathbf{r_3} = 3 \cdot (2\hat{i} - \hat{j} - 2\hat{k}) = 6\hat{i} - 3\hat{j} - 6\hat{k} \]

Now, we can add these vectors together:
\[ m_1 \mathbf{r_1} + m_2 \mathbf{r_2} + m_3 \mathbf{r_3} = (\hat{i} + 4\hat{j} + \hat{k}) + (2\hat{i} + 2\hat{j} + 2\hat{k}) + (6\hat{i} - 3\hat{j} - 6\hat{k}) \]

Combining the components:
- \( \hat{i} \): \( 1 + 2 + 6 = 9 \)
- \( \hat{j} \): \( 4 + 2 - 3 = 3 \)
- \( \hat{k} \): \( 1 + 2 - 6 = -3 \)

So, the numerator becomes:
\[ 9\hat{i} + 3\hat{j} - 3\hat{k} \]

### Step 5: Calculate the position vector of the center of mass
Now we substitute the numerator and the total mass into the formula:
\[ \mathbf{r_{cm}} = \frac{9\hat{i} + 3\hat{j} - 3\hat{k}}{6} \]

This simplifies to:
\[ \mathbf{r_{cm}} = \frac{9}{6}\hat{i} + \frac{3}{6}\hat{j} - \frac{3}{6}\hat{k} = \frac{3}{2}\hat{i} + \frac{1}{2}\hat{j} - \frac{1}{2}\hat{k} \]

### Final Answer
Thus, the position vector of the center of mass is:
\[ \mathbf{r_{cm}} = \frac{3}{2}\hat{i} + \frac{1}{2}\hat{j} - \frac{1}{2}\hat{k} \, \text{m} \]

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