Orthogonal Projections

Orthogonal projection of a point P onto a line r is the foot of the perpendicular from the point to the line.

For instance, consider a point P and a line r.

The orthogonal projection is the point P', where the perpendicular from point P intersects the line r.

Thus, in this example, the orthogonal projection is the point P' on the line r.

Through point P, there is exactly one line that is perpendicular to a given line. Therefore, it is possible to draw only one line perpendicular to line r that passes through point P.

A perpendicular is a line or segment that passes through a point P and intersects line r at a right angle (90°). In a plane, it is the shortest segment connecting point P to a point on the line.

The orthogonal projection of a segment follows a similar process.

For example, consider a segment AB and a line r.

Project the endpoints A and B of the segment onto the line and connect them.

Therefore, the orthogonal projection of segment AB onto line r is the segment A'B'.

Note: The orthogonal projection A'B' of a segment onto a line is congruent to the original segment AB only when the segment is parallel to the line.

In all other cases, the length of the orthogonal projection is always shorter than that of the original segment. $$ \overline{A'B'} \le \overline{AB} $$ In the special case where the original segment is perpendicular to the line, the orthogonal projection reduces to a single point.

And so forth.