Two vectors `vecA=3hati+2hatj+hatk" and "vecB=5hatj-9hatj+Phatk` are perpendicular to each other. The value of 'P' is :-

Options

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

Answer

To find the value of \( P \) such that the vectors \( \vec{A} = 3\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{B} = 5\hat{i} - 9\hat{j} + P\hat{k} \) are perpendicular, we can use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Write the dot product formula**:
The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = (3\hat{i} + 2\hat{j} + \hat{k}) \cdot (5\hat{i} - 9\hat{j} + P\hat{k}) \]

2. **Calculate the dot product**:
Using the distributive property of the dot product:
\[ \vec{A} \cdot \vec{B} = 3 \cdot 5 + 2 \cdot (-9) + 1 \cdot P \]
This simplifies to:
\[ \vec{A} \cdot \vec{B} = 15 - 18 + P \]

3. **Set the dot product to zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 15 - 18 + P = 0 \]

4. **Solve for \( P \)**:
Simplifying the equation gives:
\[ -3 + P = 0 \]
Adding 3 to both sides:
\[ P = 3 \]

### Conclusion:
The value of \( P \) is \( 3 \).