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28. Determining the distance from P to l point of l and m. Then, by definition, d(P, Q) is the distance from P to Q. By the definition of perpendicular, we easily see that d(P, Q) < d(P, X) for every X on l other than Q. To determine the distance from P to l, we first rewrite the equation of l in a special form. 7) of l in some coordinate system: x = q + λa, that is, x1 = q1 + λa1 , x2 = q2 + λa2 . The line l is parallel to the span of a. By eliminating λ we obtain the following equation of l: a2 x1 − a1 x2 = a2 q1 − a1 q2 .
For every point P and every line m of the plane, there exists a unique line passing through P and parallel to m. It immediately follows from the definition of parallel that every line l is parallel to itself, l // l, and that if l // m, then m // l. The first property expresses the reflexivity of the parallel relation, the second its symmetry. The following theorem asserts its transitivity. 21. Given lines l, m, and n, if l // m and m // n, then l // n. Proof. We give a proof by contradiction. Suppose l is not parallel to n; then l and n have an intersection point P .
If we imagine a physical mirror placed perpendicularly to this page, see Fig. 2, the eye sees both the figure F and mirror l eye Fig. 2. Construction of the reflected rays using the mirror image its reflection Sl (F ). In fact, we use the virtual extension inside the mirror of the light rays through which we see the reflected image to reconstruct the real light rays. The reflection axis is also called an axis of rotation. The image that is associated to this name is the plane turning over in space, where the reflection axis is used as rotation axis.