if the vectors `P = alphahati+alphaj+3hatk` and `Q = alphahati - 2 hatj - hatk` are perpendicular to each other, then the positive value of `alpha` is

Options

A. 0 (zero)
B. 1
C. 2
D. 3

Explanation

To solve the problem, we need to determine the positive value of \( \alpha \) for which the vectors \( \mathbf{P} \) and \( \mathbf{Q} \) are perpendicular. The vectors are given as:

\[ \mathbf{P} = \alpha \hat{i} + \alpha \hat{j} + 3 \hat{k} \]
\[ \mathbf{Q} = \alpha \hat{i} - 2 \hat{j} - \hat{k} \]

### Step 1: Use the condition for perpendicular vectors
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product \( \mathbf{P} \cdot \mathbf{Q} \) and set it equal to zero:

\[ \mathbf{P} \cdot \mathbf{Q} = (\alpha \hat{i} + \alpha \hat{j} + 3 \hat{k}) \cdot (\alpha \hat{i} - 2 \hat{j} - \hat{k}) \]

### Step 2: Calculate the dot product
Using the properties of the dot product:

\[ \mathbf{P} \cdot \mathbf{Q} = \alpha \cdot \alpha + \alpha \cdot (-2) + 3 \cdot (-1) \]

This simplifies to:

\[ \mathbf{P} \cdot \mathbf{Q} = \alpha^2 - 2\alpha - 3 \]

### Step 3: Set the dot product to zero
Now, we set the dot product equal to zero:

\[ \alpha^2 - 2\alpha - 3 = 0 \]

### Step 4: Solve the quadratic equation
We can factor the quadratic equation:

\[ \alpha^2 - 3\alpha + \alpha - 3 = 0 \]

This can be factored as:

\[ (\alpha - 3)(\alpha + 1) = 0 \]

### Step 5: Find the values of \( \alpha \)
Setting each factor to zero gives us:

\[ \alpha - 3 = 0 \quad \Rightarrow \quad \alpha = 3 \]
\[ \alpha + 1 = 0 \quad \Rightarrow \quad \alpha = -1 \]

### Step 6: Determine the positive value of \( \alpha \)
Since we are looking for the positive value of \( \alpha \), we conclude:

\[ \alpha = 3 \]

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

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