Question #41633

This question tests the intersection of 3D vector geometry and complex number constraints, specifically targeting the 'Zero-Factor Cancellation Trap'.

Students often attempt to algebraically manipulate the cross products without realizing that taking the dot product with $\vec{u}$ elegantly isolates the magnitude $|\vec{v}|^2 = 244$.

A common pitfall is dividing by variable vector terms or assuming $\vec{v} \cdot \vec{n} \neq 0$, which prunes the valid geometric solution where the line is parallel to the plane ($\vec{v} \cdot \vec{n} = 0$).

Furthermore, the problem forces students to verify the zero-product property $(\vec{v} \cdot \vec{n})(\vec{v} \times \vec{u}) = \vec{0}$; they must use the complex modulus constraint ($|z|^2 = 56$) to rigorously eliminate the collinearity case ($\vec{v} \times \vec{u} = \vec{0}$), proving that the line's direction vector is indeed orthogonal to the plane's normal.