If vector ` vecP=a hati + a hatj +3hatk and vecQ=a hati -2 hatj -hatk ` are perpendicular to each other , then the positive value of a is

Options

A. 3
B. 1
C. 2
D. 0

Explanation

To solve the problem, we need to determine the positive value of \( a \) such that the vectors \( \vec{P} \) and \( \vec{Q} \) are perpendicular to each other.

### Step-by-Step Solution:

1. **Identify the vectors**:
\[ \vec{P} = a \hat{i} + a \hat{j} + 3 \hat{k} \]
\[ \vec{Q} = a \hat{i} - 2 \hat{j} - \hat{k} \]

2. **Use the condition for perpendicular vectors**:
Two vectors are perpendicular if their dot product is zero:
\[ \vec{P} \cdot \vec{Q} = 0 \]

3. **Calculate the dot product**:
\[ \vec{P} \cdot \vec{Q} = (a \hat{i} + a \hat{j} + 3 \hat{k}) \cdot (a \hat{i} - 2 \hat{j} - \hat{k}) \]
Using the properties of the dot product:
\[ = a \cdot a + a \cdot (-2) + 3 \cdot (-1) \]

4. **Simplify the dot product**:
\[ = a^2 - 2a - 3 \]

5. **Set the dot product to zero**:
\[ a^2 - 2a - 3 = 0 \]

6. **Factor the quadratic equation**:
\[ (a - 3)(a + 1) = 0 \]

7. **Find the values of \( a \)**:
Setting each factor to zero gives:
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

8. **Select the positive value**:
The positive value of \( a \) is:
\[ a = 3 \]

### Final Answer:
The positive value of \( a \) is \( \boxed{3} \).