Question #40511

Let $z_1, z_2, z_3$ be the distinct complex roots of the equation $z^3 = 8$. Let $A = \begin{pmatrix} t & t^2-3 \\ 1 & t-4 \end{pmatrix}$ be a transformation matrix, where $t \in \mathbb{Z}$. For each $k \in \{1, 2, 3\}$, let $w_k = x_k + i y_k$ be the complex number defined by the matrix multiplication $\begin{pmatrix} x_k \\ y_k \end{pmatrix} = A \begin{pmatrix} \text{Re}(z_k) \\ \text{Im}(z_k) \end{pmatrix}$ Consider the vectors $\vec{v}_k = \begin{pmatrix} x_k^2 \\ y_k^2 \\ 1 \end{pmatrix}$ in $\mathbb{R}^3$. Given that the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ are linearly dependent and the points $w_1, w_2, w_3$ are not collinear, let $\Delta$ be the area of the triangle formed by $w_1, w_2, w_3$ in the complex plane. Find the value of $\Delta^2$.