A line passes through the points whose position vectors are `hati+hatj-2hatk` and `hati-3hatj+hatk`. The position vector of a point on it at unit distance from the first point is

Options

A. `1/5(5hatihatj-7hatk)`
B. `1/5(4hati+9hatj-15hatk)`
C. `(hati-4hatj+3hatk)`
D. `1/5(hati-4hatj+3hatk)`

Explanation

To find the position vector of a point on the line at unit distance from the first point, we will follow these steps:

### Step 1: Identify the position vectors of the given points
Let the position vectors of the points be:
- Point A: \(\mathbf{a} = \hat{i} + \hat{j} - 2\hat{k}\)
- Point B: \(\mathbf{b} = \hat{i} - 3\hat{j} + \hat{k}\)

### Step 2: Find the vector \( \mathbf{AB} \)
The vector \( \mathbf{AB} \) can be calculated as:
\[ \mathbf{AB} = \mathbf{b} - \mathbf{a} = (\hat{i} - 3\hat{j} + \hat{k}) - (\hat{i} + \hat{j} - 2\hat{k}) \]
\[ \mathbf{AB} = \hat{i} - 3\hat{j} + \hat{k} - \hat{i} - \hat{j} + 2\hat{k} \]
\[ \mathbf{AB} = -4\hat{j} + 3\hat{k} \]

### Step 3: Calculate the magnitude of \( \mathbf{AB} \)
The magnitude of \( \mathbf{AB} \) is given by:
\[ |\mathbf{AB}| = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 4: Find the unit vector in the direction of \( \mathbf{AB} \)
The unit vector \( \mathbf{u} \) in the direction of \( \mathbf{AB} \) is:
\[ \mathbf{u} = \frac{\mathbf{AB}}{|\mathbf{AB}|} = \frac{-4\hat{j} + 3\hat{k}}{5} = -\frac{4}{5}\hat{j} + \frac{3}{5}\hat{k} \]

### Step 5: Find the position vector of point C at unit distance from point A
To find the position vector of point C, which is at unit distance from point A in the direction of \( \mathbf{u} \):
\[ \mathbf{c} = \mathbf{a} + \mathbf{u} \]
Substituting the values:
\[ \mathbf{c} = (\hat{i} + \hat{j} - 2\hat{k}) + \left(-\frac{4}{5}\hat{j} + \frac{3}{5}\hat{k}\right) \]
\[ \mathbf{c} = \hat{i} + \left(1 - \frac{4}{5}\right)\hat{j} + \left(-2 + \frac{3}{5}\right)\hat{k} \]
\[ \mathbf{c} = \hat{i} + \frac{1}{5}\hat{j} - \frac{7}{5}\hat{k} \]

### Final Answer
The position vector of a point on the line at unit distance from the first point is:
\[ \mathbf{c} = \hat{i} + \frac{1}{5}\hat{j} - \frac{7}{5}\hat{k} \]