Question

If vectors \( 2\hat{i} + \hat{j} + \hat{k} \) and \( \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular to each other, then find the value of \( \lambda \).

Hint:

To check if two vectors are perpendicular, calculate their dot product and set it equal to zero.

Answer 1

For two vectors to be perpendicular, their dot product must be zero. The dot product of the given vectors is: \[ (2\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - 4\hat{j} + \lambda \hat{k}) = 0 \] Now, compute the dot product: \[ (2\hat{i} \cdot \hat{i}) + (\hat{j} \cdot (-4\hat{j})) + (\hat{k} \cdot (\lambda \hat{k})) = 0 \] Since \( \hat{i} \cdot \hat{i} = 1 \), \( \hat{j} \cdot \hat{j} = 1 \), and \( \hat{k} \cdot \hat{k} = 1 \), we get: \[ 2(1) + 1(-4) + \lambda(1) = 0 \] Simplifying: \[ 2 - 4 + \lambda = 0 $\Rightarrow$ \lambda = 2 \] Final Answer: The value of \( \lambda \) is \( \boxed{2} \).