A student claims that the perpendicular distance from a point to a line equals the magnitude of the projection of the joining vector onto the line's direction vector. Why is this incorrect?

Underlying idea: split the vector from a point on the line to the given point into parallel and perpendicular components relative to the line. 1. Let $\vec v$ be the vector from a point on the line to the external point, and let $\vec u$ be a direction vector of the line. 2. The projection of $\vec v$ onto $\vec u$ measures the part of $\vec v$ along the line, not away from it. 3. The shortest distance to the line is the length of the component of $\vec v$ perpendicular to $\vec u$. 4. Hence the correct distance is obtained by removing the parallel component from $\vec v$, giving $$ \text{distance}=\sqrt{|\vec v|^2-\frac{(\vec v\cdot\vec u)^2}{|\vec u|^2}}. $$ Option A reverses the roles of the components. Option C is false except when the point lies on the line. Option D is false because vector projection works in any dimension.