If the position vectors of A,B,C and D are `2hati+hatj,hati-3hatj,3hati+2hatj and hati+lamda hatj` respectively and `vec(AB) || vec(CD)`. Then `lamda` will be

Options

A. `-8`
B. `-6`
C. 8
D. 6

Explanation

To solve the problem, we need to find the value of \(\lambda\) given that the position vectors of points A, B, C, and D are as follows:

- Position vector of A: \(\vec{A} = 2\hat{i} + \hat{j}\)
- Position vector of B: \(\vec{B} = \hat{i} - 3\hat{j}\)
- Position vector of C: \(\vec{C} = 3\hat{i} + 2\hat{j}\)
- Position vector of D: \(\vec{D} = \hat{i} + \lambda \hat{j}\)

We are given that \(\vec{AB} \parallel \vec{CD}\). This means that the vectors \(\vec{AB}\) and \(\vec{CD}\) must be proportional.

### Step 1: Calculate the vector \(\vec{AB}\)

The vector \(\vec{AB}\) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Substituting the position vectors:
\[ \vec{AB} = (\hat{i} - 3\hat{j}) - (2\hat{i} + \hat{j}) = \hat{i} - 3\hat{j} - 2\hat{i} - \hat{j} \]
Simplifying this:
\[ \vec{AB} = -\hat{i} - 4\hat{j} \]

### Step 2: Calculate the vector \(\vec{CD}\)

The vector \(\vec{CD}\) is given by:
\[ \vec{CD} = \vec{D} - \vec{C} \]
Substituting the position vectors:
\[ \vec{CD} = (\hat{i} + \lambda \hat{j}) - (3\hat{i} + 2\hat{j}) = \hat{i} + \lambda \hat{j} - 3\hat{i} - 2\hat{j} \]
Simplifying this:
\[ \vec{CD} = -2\hat{i} + (\lambda - 2)\hat{j} \]

### Step 3: Set up the proportion for parallel vectors

Since \(\vec{AB} \parallel \vec{CD}\), the coefficients of \(\hat{i}\) and \(\hat{j}\) in both vectors must be proportional:
\[ \frac{-1}{-2} = \frac{-4}{\lambda - 2} \]
This simplifies to:
\[ \frac{1}{2} = \frac{-4}{\lambda - 2} \]

### Step 4: Cross-multiply to solve for \(\lambda\)

Cross-multiplying gives:
\[ 1 \cdot (\lambda - 2) = 2 \cdot (-4) \]
This simplifies to:
\[ \lambda - 2 = -8 \]
Adding 2 to both sides:
\[ \lambda = -8 + 2 \]
Thus:
\[ \lambda = -6 \]

### Final Answer
The value of \(\lambda\) is \(-6\).

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