The value of `lamda` for which the vectors `3hati-6hatj+hatkand2hati-4hatj+lamdahatk` parallel, is

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(5)/(2)`
D. `(2)/(5)`

Explanation

To find the value of \(\lambda\) for which the vectors \( \vec{A} = 3\hat{i} - 6\hat{j} + \hat{k} \) and \( \vec{B} = 2\hat{i} - 4\hat{j} + \lambda\hat{k} \) are parallel, we can use the condition 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 \( \vec{A} = 3\hat{i} - 6\hat{j} + 1\hat{k} \):
- \( x_1 = 3 \)
- \( y_1 = -6 \)
- \( z_1 = 1 \)
- For vector \( \vec{B} = 2\hat{i} - 4\hat{j} + \lambda\hat{k} \):
- \( x_2 = 2 \)
- \( y_2 = -4 \)
- \( z_2 = \lambda \)

2. **Set up the ratio for parallel vectors:**
- The condition for the vectors to be parallel is:
\[ \frac{x_1}{x_2} = \frac{y_1}{y_2} = \frac{z_1}{z_2} \]
- Substituting the values:
\[ \frac{3}{2} = \frac{-6}{-4} = \frac{1}{\lambda} \]

3. **Simplify the ratios:**
- The second ratio simplifies as follows:
\[ \frac{-6}{-4} = \frac{6}{4} = \frac{3}{2} \]
- Thus, we have:
\[ \frac{3}{2} = \frac{3}{2} = \frac{1}{\lambda} \]

4. **Solve for \(\lambda\):**
- From the equation \( \frac{1}{\lambda} = \frac{3}{2} \), we can cross-multiply:
\[ 1 \cdot 2 = 3 \cdot \lambda \]
\[ 2 = 3\lambda \]
- Now, divide both sides by 3:
\[ \lambda = \frac{2}{3} \]

5. **Conclusion:**
- The value of \(\lambda\) for which the vectors are parallel is:
\[ \lambda = \frac{2}{3} \]