If the vectors `5hat(i)+2hat(j)-hat(k)` and `lambda hat(i)-hat(j)+5hat(k)` are orthogonal vectors, then the value of `lambda` is :

Options

A. `(3)/(5)`
B. `(5)/(7)`
C. `(7)/(5)`
D. `(2)/(5)`

Answer

To find the value of \( \lambda \) such that the vectors \( \mathbf{A} = 5\hat{i} + 2\hat{j} - \hat{k} \) and \( \mathbf{B} = \lambda\hat{i} - \hat{j} + 5\hat{k} \) are orthogonal, we will use the property that two vectors are orthogonal if their dot product is zero.

### Step-by-step Solution:

1. **Write the vectors**:
\[ \mathbf{A} = 5\hat{i} + 2\hat{j} - \hat{k} \]
\[ \mathbf{B} = \lambda\hat{i} - \hat{j} + 5\hat{k} \]

2. **Calculate the dot product**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = (5)(\lambda) + (2)(-1) + (-1)(5) \]

3. **Simplify the expression**:
\[ \mathbf{A} \cdot \mathbf{B} = 5\lambda - 2 - 5 \]
\[ \mathbf{A} \cdot \mathbf{B} = 5\lambda - 7 \]

4. **Set the dot product to zero** (since the vectors are orthogonal):
\[ 5\lambda - 7 = 0 \]

5. **Solve for \( \lambda \)**:
\[ 5\lambda = 7 \]
\[ \lambda = \frac{7}{5} \]

### Final Answer:
The value of \( \lambda \) is \( \frac{7}{5} \).