If the vectors `3hati+lambdahatj+hatk` and `2hati-hatj+8hatk` are perpendicular, then `lambda` is equal to

Options

A. -14
B. 7
C. 14
D. `(1)/(7)`

Answer

To solve the problem, we need to find the value of \(\lambda\) such that the vectors \( \mathbf{A} = 3\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \mathbf{B} = 2\hat{i} - \hat{j} + 8\hat{k} \) are perpendicular.

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product of the two given vectors and set it equal to zero.

2. **Write the Dot Product**:
The dot product of vectors \(\mathbf{A}\) and \(\mathbf{B}\) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = (3\hat{i} + \lambda\hat{j} + \hat{k}) \cdot (2\hat{i} - \hat{j} + 8\hat{k}) \]

3. **Calculate the Dot Product**:
Using the properties of dot products, we compute:
\[ \mathbf{A} \cdot \mathbf{B} = 3 \cdot 2 + \lambda \cdot (-1) + 1 \cdot 8 \]
\[ = 6 - \lambda + 8 \]
\[ = 14 - \lambda \]

4. **Set the Dot Product to Zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 14 - \lambda = 0 \]

5. **Solve for \(\lambda\)**:
Rearranging the equation gives:
\[ \lambda = 14 \]

### Conclusion:
The value of \(\lambda\) is \(14\).