Question #41625

This question tests the student's ability to correctly identify the direction ratios of an orthogonally projected line.

A common 'parametric form pitfall' occurs when students incorrectly assume that the orthogonal projection of a line onto a plane merely translates the line, preserving its original direction vector $\vec{b}$.

This happens when they treat the parameter $t$ as a constant during the projection process rather than a varying scalar, leading to a failure in applying the projection operator $P = I - \frac{\vec{n}\vec{n}^T}{|\vec{n}|^2}$.

By using the original direction ratios $\langle 1, -1, 2 \rangle$ instead of the projected ones $\langle 5, -1, 4 \rangle$, they arrive at the trap answer 9.

Another trap (yielding 10) catches students who project a single point correctly but still reuse the original direction vector.