Parallelogram area with the edge directions fixed

Problem

On the coordinate plane, one pair of sides of a parallelogram is parallel to $5x-y=0$, and the other pair lies on lines perpendicular to $3x-2y=0$. The diagonals meet at $Q$, and one vertex $P$ satisfies $\overrightarrow{PQ}=(10,-1)$. Find the area. The full question is on the CEEC 114 AST Mathematics I paper.

Where it goes wrong

Writing coordinates for all four vertices makes a large system, and a sign error is easy. The edge directions are already fixed, and the vector from the center to a vertex is only half a diagonal. The area is the absolute value of the cross product of the two side vectors, so the four vertices do not all have to be written down.

Idea

Take ${\mathbf{u}}_0=(1,5)$, parallel to $5x-y=0$. Lines perpendicular to $3x-2y=0$ have slope $-2/3$, so take ${\mathbf{v}}_0=(3,-2)$. If the side vectors are $\alpha {\mathbf{u}}_0$ and $\beta {\mathbf{v}}_0$, then

All four solutions have $|\alpha \beta |=12$, and

so the area is

What the figure shows

• The blue and purple dashed lines are the two families of direction rails. Blue is parallel to $5x-y=0$. Purple is perpendicular to $3x-2y=0$.
• Drag the tip of the blue arrow $\mathbf{u}=\alpha {\mathbf{u}}_0$ or the purple arrow $\mathbf{v}=\beta {\mathbf{v}}_0$ through $P$ to flip the sign of $\alpha$ or $\beta$. The sidebar picks the same four sign choices: Half-diagonal = ½(u+v) or Half-diagonal = ½(u−v) is the inner $\pm$, and PQ same direction or PQ reversed is the outer $\pm$.
• $Q$ stays at the origin and $P=(-10,1)$, so $\overrightarrow{PQ}=(10,-1)$ as in the paper. All four sign choices describe this same gold parallelogram; only the arrows along the rails flip.
• The drag is not continuous. With $\overrightarrow{PQ}$ fixed and both directions locked, only these four sign choices fit, and the area is always $204$ ($|\alpha \beta |=12$ times the direction cross product $17$).
• The sidebar value $|\mathbf{u}\times \mathbf{v}|$ is that area. The four vertices do not have to be solved one by one.