Closest Point 2a: Shortest Distance in 3D with Vectors and Linear Functions

This example demonstrates how to find the closest point on a line (or plane) to a given point in three-dimensional space, and compute the shortest distance using geometric construction. Core mathematical concepts include 3D vectors, linear functions, orthogonal projection, distance formulas, and relationships involving scalene triangles and rectangles. Construction details: using GeoGebra commands Line, Point, Segment, OrthogonalLine, Intersect, and Angle, with objects A, B, f, x_0, w, d, text2, h, w_0, i, α, j, k, C, d_y, d_x, s_{1}, s_{2}, s, text3, text4. The perpendicular foot is determined via orthogonal lines and intersection points, and distances are verified using angle and segment measurements. Suitable for high school or early college students learning analytic geometry, vector algebra, and introductory optimization, as well as for teachers demonstrating the intuitive concept of point-line distance in 3D.