Question #41226

This question tests the student's ability to recognize the equality condition of the Triangle Inequality in the complex plane, disguised behind an involutory matrix transformation.

The trap lies in the matrix equation $A\vec{Z} = -\vec{Z}$, which simplifies to the plane equation $z_1 + z_2 + z_3 = 0$.

Students often attempt to expand $|z_1^2 + z_2^2 + z_3^2|$ algebraically using $\sum z_i^2 = -2 \sum z_i z_j$, which leads to a dead end because the relative arguments (angles) between the complex numbers are seemingly unknown.

The 'Lethal Collision' occurs when they fail to notice that $|z_3| = |z_1| + |z_2|$ (since $15 = 7 + 8$), which geometrically forces $z_1$ and $z_2$ to be collinear.

Recognizing this degenerate locus is mandatory to collapse the complex variables into a single argument $e^{i\theta}$, allowing exact evaluation of the modulus.