A vector `P=3hati-2hatj+ahatk` is perpendicular to the vector `Q=2hati+hatj -hatk,` The value of a is

Options

A. 2
B. 1
C. 4
D. 3

Explanation

To solve the problem, we need to find the value of \( a \) such that the vector \( P = 3\hat{i} - 2\hat{j} + a\hat{k} \) is perpendicular to the vector \( Q = 2\hat{i} + \hat{j} - \hat{k} \).

### Step-by-Step Solution:

1. **Understand the Condition for Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product \( P \cdot Q \) and set it equal to zero.

2. **Write the Dot Product**:
The dot product of two vectors \( A = a_1\hat{i} + b_1\hat{j} + c_1\hat{k} \) and \( B = a_2\hat{i} + b_2\hat{j} + c_2\hat{k} \) is given by:
\[ A \cdot B = a_1a_2 + b_1b_2 + c_1c_2 \]
For our vectors \( P \) and \( Q \):
- \( P = 3\hat{i} - 2\hat{j} + a\hat{k} \)
- \( Q = 2\hat{i} + \hat{j} - \hat{k} \)

3. **Calculate the Dot Product**:
\[ P \cdot Q = (3)(2) + (-2)(1) + (a)(-1) \]
Simplifying this gives:
\[ P \cdot Q = 6 - 2 - a \]

4. **Set the Dot Product Equal to Zero**:
Since \( P \) and \( Q \) are perpendicular, we set the dot product to zero:
\[ 6 - 2 - a = 0 \]

5. **Solve for \( a \)**:
Rearranging the equation gives:
\[ 4 - a = 0 \]
Thus:
\[ a = 4 \]

### Final Answer:
The value of \( a \) is \( 4 \).

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