If , , , then is:

(d) -74 Given $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$, $\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}$, $\vec{c} = \hat{i} + \hat{j} + \hat{k}$. $\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ -1 & 2 & -4 \end{vmatrix} = \hat{i}(-12 + 2) - \hat{j}(-8 - 1) + \hat{k}(4 + 3) = -10\hat{i} + 9\hat{j} + 7\hat{k}$. $\vec{a} \times \vec{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} = \hat{i}(3 + 1) - \hat{j}(2 + 1) + \hat{k}(2 - 3) = 4\hat{i} - 3\hat{j} - \hat{k}$. Therefore, $(\vec{a} \times \vec{b}) \cdot (\vec{a} \times \vec{c}) = (-10)(4) + (9)(-3) + (7)(-1) = -40 - 27 - 7 = -74$.