Given that $\vec a = -3\hat i-6\hat j+4\hat k, \vec b = 9\hat i - λ\hat j-12\hat k$ if $\vec a×\vec b=\vec 0$, then the value of $λ$ is

-18

The correct answer is Option (1) → -18

$\vec{a} = -3\hat{i} - 6\hat{j} + 4\hat{k}$, $\vec{b} = 9\hat{i} - \lambda\hat{j} - 12\hat{k}$

$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3 & -6 & 4 \\ 9 & -\lambda & -12 \end{vmatrix} = 0$

Expansion:

$\hat{i}[(-6)(-12) - (4)(-\lambda)] - \hat{j}[(-3)(-12) - (4)(9)] + \hat{k}[(-3)(-\lambda) - (-6)(9)] = 0$

$\hat{i}[72 + 4\lambda] - \hat{j}[36 - 36] + \hat{k}[3\lambda + 54] = 0$

From $\hat{i}$-component: $72 + 4\lambda = 0 \Rightarrow \lambda = -18$

From $\hat{k}$-component: $3\lambda + 54 = 0 \Rightarrow \lambda = -18$

Hence, ${\lambda = -18}$