Question #72147

The problem tests the angle between a line and a plane, but the plane equation is given as a determinant.

The primary trap is the 'Minors and Cofactors' sign convention.

If a student expands the determinant along the second row but forgets the alternating signs $(-1)^{i+j}$ for cofactors $C_{21}$ and $C_{23}$, they obtain an incorrect plane equation ($x + 2y - z - 6 = 0$) with normal $\langle 1, 2, -1 \rangle$.

The problem forces a Coplanarity Condition check: the line $L$ is parallel to both the true and wrong planes, so the student must substitute a point to find $\lambda$.

The wrong plane yields $\lambda = 0$, which perfectly cascades into the second part, making the direction vector of $L'$ orthogonal to the wrong normal, resulting in a trap answer of $\theta = 0$.

The correct expansion yields $\lambda = 4$ and $\theta = \pi/6$.