The value of `lamda` for which the vectors `3hat(i)-6hat(j)+hat(k)` and `2hat(i)-4hat(j)+lamda hat(k)` are parallel is

Options

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

Answer

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 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 \( \vec{A} \):
- \( x_1 = 3 \)
- \( y_1 = -6 \)
- \( z_1 = 1 \)
- For vector \( \vec{B} \):
- \( x_2 = 2 \)
- \( y_2 = -4 \)
- \( z_2 = \lambda \)

2. **Set up the ratios for parallel vectors:**
Since the vectors are parallel, we can set up the following equations based on the ratios of their components:
\[ \frac{x_1}{x_2} = \frac{y_1}{y_2} = \frac{z_1}{z_2} \]
This gives us:
\[ \frac{3}{2} = \frac{-6}{-4} = \frac{1}{\lambda} \]

3. **Simplify the ratios:**
- The second ratio simplifies to:
\[ \frac{-6}{-4} = \frac{6}{4} = \frac{3}{2} \]
Thus, we have:
\[ \frac{3}{2} = \frac{3}{2} \]
This confirms that the first two ratios are equal.

4. **Solve for \(\lambda\):**
Now, we can use the first and the third ratio:
\[ \frac{3}{2} = \frac{1}{\lambda} \]
Cross-multiplying gives:
\[ 3\lambda = 2 \]
Dividing both sides by 3:
\[ \lambda = \frac{2}{3} \]

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