If the planes `overset(to)( r) (2 hat(i) - lambda (j) + 3 hat(k) ) = 0 and overset(to)( r) ( lambda (i) + 5 hat(j) - hat(k) )=5` are perpendicular to each other, then the value of `lambda` is

Options

A. `0`
B. `-2`
C. `-1`
D. `2`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the given planes are perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the Normal Vectors of the Planes**:
The equations of the planes are given in vector form. The normal vector of the first plane can be identified from the equation:
\[ \vec{r} \cdot (2 \hat{i} - \lambda \hat{j} + 3 \hat{k}) = 0 \]
This means the normal vector \( \vec{a} \) of the first plane is:
\[ \vec{a} = 2 \hat{i} - \lambda \hat{j} + 3 \hat{k} \]

The second plane is given by:
\[ \vec{r} \cdot (\lambda \hat{i} + 5 \hat{j} - \hat{k}) = 5 \]
So, the normal vector \( \vec{b} \) of the second plane is:
\[ \vec{b} = \lambda \hat{i} + 5 \hat{j} - \hat{k} \]

2. **Condition for Perpendicularity**:
Two planes are perpendicular if their normal vectors are perpendicular. This condition can be expressed mathematically as:
\[ \vec{a} \cdot \vec{b} = 0 \]

3. **Calculate the Dot Product**:
Now, we calculate the dot product \( \vec{a} \cdot \vec{b} \):
\[ \vec{a} \cdot \vec{b} = (2 \hat{i} - \lambda \hat{j} + 3 \hat{k}) \cdot (\lambda \hat{i} + 5 \hat{j} - \hat{k}) \]
Expanding this, we get:
\[ = 2\lambda + (-\lambda)(5) + 3(-1) \]
Simplifying this, we have:
\[ = 2\lambda - 5\lambda - 3 \]
\[ = -3\lambda - 3 \]

4. **Set the Dot Product to Zero**:
For the planes to be perpendicular, we set the dot product equal to zero:
\[ -3\lambda - 3 = 0 \]

5. **Solve for \( \lambda \)**:
Rearranging the equation gives:
\[ -3\lambda = 3 \]
Dividing both sides by -3:
\[ \lambda = -1 \]

### Conclusion:
The value of \( \lambda \) is \( -1 \).