Question

If \( \mathbf{A} = \hat{i} + 12 \hat{j} + \hat{k} \) and \( \mathbf{B} = 2\hat{i} - a\hat{j} + \hat{k} \) are perpendicular to each other, then what will be the value of \( a \)?

• A. 0.5
• B. -0.5
• C. 1
• D. -1

Hint:

When vectors are perpendicular, their dot product is zero. Use this property to find unknown components in vector problems.

Answer 1

The Correct Option is A

Step 1: Using the condition for perpendicular vectors.
For two vectors \( \mathbf{A} \) and \( \mathbf{B} \) to be perpendicular, their dot product must be zero: \[ \mathbf{A} \cdot \mathbf{B} = 0 \] Step 2: Finding the dot product.
The dot product of \( \mathbf{A} = \hat{i} + 12 \hat{j} + \hat{k} \) and \( \mathbf{B} = 2\hat{i} - a\hat{j} + \hat{k} \) is: \[ \mathbf{A} \cdot \mathbf{B} = (1)(2) + (12)(-a) + (1)(1) \] \[ \mathbf{A} \cdot \mathbf{B} = 2 - 12a + 1 = 0 \] Step 3: Solving for \( a \).
Simplifying: \[ 3 - 12a = 0 \] \[ 12a = 3 \] \[ a = \frac{3}{12} = 0.5 \] Thus, the correct answer is (A) 0.5.