The two vectors `veca =2hati+hatj+3hatk and vecb =4hati-lamdahatj+6hatk` are parallel, if `lamda` is equal to: a) 2 b) -3 c) 3 d) -2

Options

A. 2
B. `-3`
C. 3
D. `-2`

Answer

To determine the value of \(\lambda\) for which the vectors \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\) and \(\vec{b} = 4\hat{i} - \lambda\hat{j} + 6\hat{k}\) 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. **Write down the vectors**:
\[ \vec{a} = 2\hat{i} + \hat{j} + 3\hat{k} \]
\[ \vec{b} = 4\hat{i} - \lambda\hat{j} + 6\hat{k} \]

2. **Set up the ratio condition for parallel vectors**:
For two vectors \(\vec{a}\) and \(\vec{b}\) to be parallel, the following condition must hold:
\[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} \]
Here, \(a_1 = 2\), \(b_1 = 4\), \(a_2 = 1\), \(b_2 = -\lambda\), \(a_3 = 3\), and \(b_3 = 6\).

3. **Set up the equations from the ratios**:
From the first two components:
\[ \frac{2}{4} = \frac{1}{-\lambda} \]
From the last two components:
\[ \frac{3}{6} = \frac{1}{-\lambda} \]

4. **Simplify the ratios**:
The first ratio simplifies to:
\[ \frac{1}{2} = \frac{1}{-\lambda} \]
Cross-multiplying gives:
\[ -\lambda = 2 \]
Thus, we have:
\[ \lambda = -2 \]

5. **Verify with the second ratio**:
The second ratio simplifies to:
\[ \frac{1}{2} = \frac{1}{-\lambda} \]
Again, cross-multiplying gives:
\[ -\lambda = 2 \]
This confirms that \(\lambda = -2\).

### Final Answer:
\(\lambda = -2\)