Multi-Variable Calculus Assignment

due Friday, February 2, 2001

Problems

1. 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 Delta be the triangle with these points as vertices. Find:

   1. The parametric representation of the line through B and C that assigns parameter value 0 to the point B and parameter value 1 to the point C.

   2. The midpoint M of the line segment BC.

   3. The point on BC that is one-third of the way from B to C.

   4. The parametric representation of the line through A and M that assigns parameter value 0 to A and parameter value 1 to M.

   5. Recall from plane geometry where the median of Delta is located on AM, and then determine that point.

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

   L: F(t) = (3t, -t, t + 2) M: G(t) = (2t - 1, -t + 1, t + 1) .

   1. Determine the points a on L and b on M obtained from parameter value 0 using F and G, respectively.

   2. Determine the vector v along L and w along M that runs, in each case, from the point where t = 0 to the point where t = 1.

   3. Find the angle between the vectors v and w.

   4. Find the ``cross product'' u = v \times w.

   5. Compute u . F(t) and u . G(t).

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