Question

The vector \(\vec{P}=a\hat{i}+a\hat{j}+3\hat{k}\) and \(\vec{Q}=a\hat{i}-2\hat{j}-\hat{k}\) are perpendicular to each other. Then positive value of '\(a\)' is

• A. \(3\)
• B. \(4\)
• C. \(9\)
• D. \(13\)

Hint:

Two vectors are perpendicular when their dot product is zero. Set up the equation in a and take the positive root.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Two vectors are perpendicular when their scalar (dot) product is zero.

Step 2: Dot Product
\(\vec{P}\cdot\vec{Q}=a\cdot a+a\cdot(-2)+3\cdot(-1)=a^{2}-2a-3\).

Step 3: Solve the Equation
Put \(a^{2}-2a-3=0\), so \((a-3)(a+1)=0\) and \(a=3\) or \(a=-1\).

Step 4: Choose the Positive Value
The question asks for the positive value, so \(a=3\).

Final Answer:
The positive value of a is 3. \[ \boxed{\text{(A) }3} \]