Question #75847

Let $\vec{u}_1 = (\{x\}, \{y\}, \{z\})$, $\vec{u}_2 = (\{y\}, \{z\}, \{x\})$, $\vec{u}_3 = (\{z\}, \{x\}, \{y\})$ and $\vec{v}_1 = ([x], [y], [z])$, $\vec{v}_2 = ([y], [z], [x])$, $\vec{v}_3 = ([z], [x], [y])$. Let $\vec{w}_1 = \hat{i}, \vec{w}_2 = \hat{j}, \vec{w}_3 = \hat{k}$ and $\vec{x}_1 = \hat{j}, \vec{x}_2 = \hat{k}, \vec{x}_3 = \hat{i}$. A $3 \times 3$ matrix $M$ is defined by its entries $M_{ij} = (\vec{u}_i \cdot \vec{w}_j)(\vec{v}_i \cdot \vec{x}_j) - (\vec{u}_i \cdot \vec{x}_j)(\vec{v}_i \cdot \vec{w}_j)$. If $x, y, z$ form an arithmetic progression with a non-zero common difference, and $[x], [y], [z]$ also form an arithmetic progression, find $\det(M)$.