Question #41649

The core trap lies in the standard parameterization of a family of planes.

Students habitually use $P_1 + \lambda P_2 = 0$ to represent any plane passing through the intersection of $P_1$ and $P_2$.

However, this asymmetric form inherently excludes the plane $P_2$ itself (which corresponds to $\lambda \to \infty$).

When students apply the parallelism condition $(\vec{n}_1 + \lambda \vec{n}_2) \cdot \vec{d} = 0$, they obtain a contradictory equation ($-1 = 0$), leading them to falsely conclude that no such plane exists or to guess incorrectly.

By using the homogeneous symmetric form $\alpha P_1 + \beta P_2 = 0$, the linear dependence is properly handled, revealing that the required plane is exactly $P_2$.