Let's break down the question step by step.

We are given the following vectors:

$\vec{a} = \sin\theta \hat{i} + \cos\theta \hat{j} + 0\hat{k}$ $\vec{b} = \cos\theta \hat{i} - \sin\theta \hat{j} - 3\hat{k}$ $\vec{c} = 2\hat{i} + 3\hat{j} - 3\hat{k}$

And we are tasked with finding: $\frac{d}{d\theta} \left[ \vec{a} \times (\vec{b} \times \vec{c}) \right] \text{ at } \theta = \frac{\pi}{2}$

Step 1: Compute $\vec{b} \times \vec{c}$

We need to calculate the cross product $\vec{b} \times \vec{c}$. The cross product between two vectors $\vec{b}$ and $\vec{c}$ in component form can be computed using the determinant of a matrix: