A line with direction ratio `(2,1,2)` intersects the lines `vecr=-hatj+lambda(hati+hatj+hatk)` and `vecr=-hati+mu(2hati+hatj+hatk)` at A and B, respectively then length of AB is equal to

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the length of the line segment \( AB \) where line \( A \) intersects with line 1 and line \( B \) intersects with line 2, we will follow these steps:

### Step 1: Write the equations of the lines

The equations of the lines can be expressed in parametric form based on the given direction ratios and points of intersection.

**Line 1**:
Given the equation \( \vec{r} = -\hat{j} + \lambda (\hat{i} + \hat{j} + \hat{k}) \), we can express it as:
\[ \vec{r_1} = (0 + \lambda) \hat{i} + (-1 + \lambda) \hat{j} + (0 + \lambda) \hat{k} \]
Thus, the coordinates of point \( A \) are:
\[ A(\lambda, \lambda - 1, \lambda) \]

**Line 2**:
Given the equation \( \vec{r} = -\hat{i} + \mu (2\hat{i} + \hat{j} + \hat{k}) \), we can express it as:
\[ \vec{r_2} = (-1 + 2\mu) \hat{i} + (0 + \mu) \hat{j} + (0 + \mu) \hat{k} \]
Thus, the coordinates of point \( B \) are:
\[ B(-1 + 2\mu, \mu, \mu) \]

### Step 2: Set up the equations based on direction ratios

The direction ratios of the line that intersects both \( A \) and \( B \) are given as \( (2, 1, 2) \). Therefore, we can set up the following equations based on the direction ratios:

1. \(\frac{2\mu - 1 - \lambda}{2} = \frac{\mu - \lambda + 1}{1} = \frac{\mu - \lambda - \lambda}{2}\)

### Step 3: Solve the equations

From the first two terms:
\[ 2\mu - 1 - \lambda = 2(\mu - \lambda + 1) \]
Expanding this gives:
\[ 2\mu - 1 - \lambda = 2\mu - 2\lambda + 2 \]
Rearranging gives:
\[ \lambda = 3 \]

Now substituting \( \lambda = 3 \) into the second equation:
\[ \frac{\mu - 3 + 1}{1} = \frac{\mu - 3}{2} \]
Cross-multiplying gives:
\[ 2(\mu - 2) = \mu - 3 \]
This simplifies to:
\[ 2\mu - 4 = \mu - 3 \implies \mu = 1 \]

### Step 4: Find the coordinates of points \( A \) and \( B \)

Substituting \( \lambda = 3 \) into the coordinates of point \( A \):
\[ A(3, 3 - 1, 3) = A(3, 2, 3) \]

Substituting \( \mu = 1 \) into the coordinates of point \( B \):
\[ B(-1 + 2 \cdot 1, 1, 1) = B(1, 1, 1) \]

### Step 5: Calculate the length of segment \( AB \)

Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of \( A(3, 2, 3) \) and \( B(1, 1, 1) \):
\[ AB = \sqrt{(1 - 3)^2 + (1 - 2)^2 + (1 - 3)^2} \]
Calculating this gives:
\[ AB = \sqrt{(-2)^2 + (-1)^2 + (-2)^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Final Answer
The length of segment \( AB \) is \( 3 \).

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