Find the position vector of a point R which divides the line joining the point `P(hati + 2hatj - hatk)` and `Q(-hati + hatj + hatk)` in the ratio `2 : 1` internally .

Answer

To find the position vector of point R that divides the line segment joining points P and Q in the ratio 2:1 internally, we can use the section formula.

### Given:
- Point P: \( \mathbf{P} = \hat{i} + 2\hat{j} - \hat{k} \)
- Point Q: \( \mathbf{Q} = -\hat{i} + \hat{j} + \hat{k} \)
- Ratio: \( 2:1 \)

### Step 1: Identify the position vectors
The position vectors of points P and Q can be expressed as:
- \( \mathbf{P} = \hat{i} + 2\hat{j} - \hat{k} \)
- \( \mathbf{Q} = -\hat{i} + \hat{j} + \hat{k} \)

### Step 2: Apply the section formula
The position vector \( \mathbf{R} \) that divides the line segment joining \( \mathbf{P} \) and \( \mathbf{Q} \) in the ratio \( m:n \) (where \( m = 2 \) and \( n = 1 \)) is given by the formula:
\[ \mathbf{R} = \frac{m\mathbf{Q} + n\mathbf{P}}{m+n} \]

Substituting the values:
\[ \mathbf{R} = \frac{2\mathbf{Q} + 1\mathbf{P}}{2+1} \]

### Step 3: Substitute the position vectors
Now, substituting the vectors:
\[ \mathbf{R} = \frac{2(-\hat{i} + \hat{j} + \hat{k}) + 1(\hat{i} + 2\hat{j} - \hat{k})}{3} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ = 2(-\hat{i}) + 2(\hat{j}) + 2(\hat{k}) + \hat{i} + 2\hat{j} - \hat{k} \]
\[ = -2\hat{i} + 2\hat{j} + 2\hat{k} + \hat{i} + 2\hat{j} - \hat{k} \]
Combining like terms:
\[ = (-2 + 1)\hat{i} + (2 + 2)\hat{j} + (2 - 1)\hat{k} \]
\[ = -\hat{i} + 4\hat{j} + \hat{k} \]

### Step 5: Divide by the total ratio
Now, we divide by 3:
\[ \mathbf{R} = \frac{-\hat{i} + 4\hat{j} + \hat{k}}{3} \]
\[ = -\frac{1}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{1}{3}\hat{k} \]

### Final Answer
Thus, the position vector of point R is:
\[ \mathbf{R} = -\frac{1}{3}\hat{i} + \frac{4}{3}\hat{j} + \frac{1}{3}\hat{k} \]

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