Question

(c) If the vectors \( \vec{v_1} = 3\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{v_2} = \hat{i} - 4\hat{j} + \lambda \hat{k} \) are perpendicular, find the value of \( \lambda \):

Hint:

For perpendicular vectors, their dot product is always zero.

Answer 1

Two vectors are perpendicular if their dot product is zero: \[ \vec{v_1} \cdot \vec{v_2} = 0. \] Substituting the given vectors: \[ (3)(1) + (2)(-4) + (1)(\lambda) = 0. \] \[ 3 - 8 + \lambda = 0 \quad \Rightarrow \quad \lambda = 5. \]