Multivariable Calculus (Math 214) Assignment

Reminder: Midterm Test Wednesday, October 20

Assignment due Friday, October 8

1. Let E_{1} , E_{2} , and E_{3} be the unit points on the positive coordinate axes in space, that is,

   E_{1} = (1,0,0) , E_{2} = (0,1,0) , E_{3} = (0,0,1) .

   The unique plane Pi through these three points may be represented as the set of solutions of the equation

   x + y + z = 1 .

   1. Find all vectors that are normal to Pi.

   2. Find two vectors that are parallel to (line segments in) Pi.

   3. Give a parametric representation of the plane Pi.

2. Projection of one vector on another. Let V be any vector in space and U any non-zero vector in space. Show that there is one and only one scalar t such that V = t U + W, where W is perpendicular to U. The vector t U is the parallel projection of V on U, while the vector W is the perpendicular projection of V on U.

3. Let A , B , and C be the points

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

   1. Find the perpendicular projection of the vector from A to C on the vector from A to B.

   2. Find the distance from C to the line through A and B.

   3. Find the equation of the plane containing the three points A , B , and C .

   4. Find the equation of the plane containing A and B that is perpendicular to the plane containing all three of the points.

4. Describe a general method for finding a vector that is normal to the unique plane determined by three non-collinear points.

5. Find the radii of the circumscribed and inscribed circles for the equilateral triangle with vertices the three points E_{1}, E_{2}, E_{3} of exercise 1.