- 5 TRANSFORMATIONS 301
S that is symmetric to A k with respect to P. (Note that Bk = Aj for some j.) By the
definition of symmetry, we have
Summing this from k = 1 to k = n implies that
By the same reasoning, we also have
� --:----t �-
QAt +QA 2 +"'+QAn = 0.Subtracting, we get nQP = 0, so Q = P. •
Besides adding, subtracting, and multiplying vectors by scalars, we can "multiply"
two vectors with the dot-product operation. Recall the formula A .8/ (IAI· 18 1) = cos e,
where I X I denotes the magnitude of the vector X and e is the angle needed to rotate A
so that it is parallel with 8 (i.e., the "angle between" the two vectors). The dot-product
can be easily computed using coordinates, but coordinates should be avoided when
possible. Three things make dot products occasionally helpful in Euclidean geometry
problems.
- Dot product is commutative and algebraically acts like ordinary multiplication
in that it obeys a "distributive law:" - The dot product of a vector with itself equals the square of the vector's magni
tude: AB. AB = AB^2. - A.-l 8 if and only if A. 8 = O.
You may want to apply these ideas to Problems 8.5.24-8.5.26.Reflections and Glide Reflections
A reflection across a line 1! maps each point X to a point X' such that 1! is the perpen
dicular bisector of XX'. You are an old hand at reflections from Section 3.1. Notice
that each point in 1! is invariant with respect to reflection across 1!; this contrasts with
translations, which have no invariant points (unless the translation is the zero vector,
i.e., the identity transformation, which fixes every point).
A glide reflection along a line 1! is a composition of two maps: first a reflection
across 1!, followed by a translation parallel to 1!. Unlike reflections, a glide reflection
has no fixed points. However, the entire line 1! is fixed; in other words, every point in 1!
gets mapped to another point in 1!.