A triple of numbers may be used to represent a point of R^{3}. If u = (u_{1}, u_{2}, u_{3}) is a triple, then the function T_{u} defined for x = (x_{1}, x_{2}, x_{3}) by T_{u}(x) = x + u is called translation by u. For two given points a and b there is one and only one translation T for which T(a) = b, namely T = T_{b - a}. We use the word vector for a triple of numbers that is being regarded as the triple associated to a translation. Sometimes b - a will, with a certain looseness of language, be called the vector from a to b. If u is a vector, sometimes the directed line segment from a point a to the point b = T_{u}(a) = a + u will be called ``u based at a''. Thus, the angle between two vectors refers to the angle formed when the vectors are based at the origin.
Two lines L and M in space are parameterized, respectively, by functions F and G that are given by the formulas
Find the point in R^{3} where these lines meet, and find the smaller of the two angles between them at that point.
Let A, B, and C be the points in R^{3} given by
and let Delta be the triangle with these points as vertices. ``Solve'' this triangle. That is: find the lengths of each of the three sides, and find the cosines of the angles in Delta at each of the three vertices.
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