Multi-Variable Calculus Assignment

due Tuesday, January 30, 2001

1 Triples, Points, and Translations

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.

2 Problems

1. Two lines L and M in space are parameterized, respectively, by functions F and G that are given by the formulas

F (t) = (3t, −t, t + 2) G(t) = (2t − 1, −t + 1, t + 1) .

Find the point in R^{3} where these lines meet, and find the smaller of the two angles between them at that point.

2. Let A, B, and C be the points in R^{3} given by

A = (2, −3, 5) , B = (−1, 0, −1) , C = (3, 1, 4) ,

and let Δ 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 Δ at each of the three vertices.

[Processed from GELLMU to L^{A}TEX on Fri 26 Jan 2001 at 08:50:33 EST]