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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.