If two vectors `2hati+3hatj-hatk` and `-4hati-6hatj-lamdahatk` are parallel to each other then value of `lamda` be

Options

A. 0
B. 2
C. 3
D. 4

Explanation

To find the value of \( \lambda \) such that the vectors \( \mathbf{A} = 2\hat{i} + 3\hat{j} - \hat{k} \) and \( \mathbf{B} = -4\hat{i} - 6\hat{j} - \lambda\hat{k} \) are parallel, we can use the property that two vectors are parallel if the ratios of their corresponding components are equal.

### Step-by-Step Solution:

1. **Identify the components of the vectors**:
- For vector \( \mathbf{A} \):
- \( A_x = 2 \) (coefficient of \( \hat{i} \))
- \( A_y = 3 \) (coefficient of \( \hat{j} \))
- \( A_z = -1 \) (coefficient of \( \hat{k} \))
- For vector \( \mathbf{B} \):
- \( B_x = -4 \) (coefficient of \( \hat{i} \))
- \( B_y = -6 \) (coefficient of \( \hat{j} \))
- \( B_z = -\lambda \) (coefficient of \( \hat{k} \))

2. **Set up the ratios**:
Since the vectors are parallel, we can write:
\[ \frac{A_x}{B_x} = \frac{A_y}{B_y} = \frac{A_z}{B_z} \]
Substituting the values:
\[ \frac{2}{-4} = \frac{3}{-6} = \frac{-1}{-\lambda} \]

3. **Calculate the first ratio**:
\[ \frac{2}{-4} = -\frac{1}{2} \]

4. **Calculate the second ratio**:
\[ \frac{3}{-6} = -\frac{1}{2} \]

5. **Set the third ratio equal to the others**:
\[ \frac{-1}{-\lambda} = -\frac{1}{2} \]
This simplifies to:
\[ \frac{1}{\lambda} = \frac{1}{2} \]

6. **Cross-multiply to solve for \( \lambda \)**:
\[ 1 \cdot 2 = 1 \cdot \lambda \]
Thus, we find:
\[ \lambda = 2 \]

### Conclusion:
The value of \( \lambda \) such that the vectors are parallel is \( \lambda = 2 \).