Question

For which value of \( \lambda \) are the vectors \( \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + 2\hat{k} \) perpendicular?

• A. -2
• B. 2
• C. 3
• D. 4

Hint:

To determine if two vectors are perpendicular, calculate their dot product. If the dot product equals zero, the vectors are perpendicular.

Answer 1

The Correct Option is B

Step 1: Use the condition for perpendicular vectors.
Two vectors are perpendicular if their dot product is zero. The dot product of the vectors \( \lambda \hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + 2\hat{k} \) is given by: \[ (\lambda \hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - 4\hat{j} + 2\hat{k}) \] Step 2: Calculate the dot product. \[ = \lambda(1) + (1)(-4) + (1)(2) = \lambda - 4 + 2 = \lambda - 2 \] Step 3: Set the dot product equal to zero for perpendicular vectors.
For the vectors to be perpendicular, the dot product must be zero: \[ \lambda - 2 = 0 \quad \Rightarrow \quad \lambda = 2 \] Thus, the correct value of \( \lambda \) is \( 2 \), and the correct answer is (B) 2.