Multi-Variable Calculus Assignment

due Tuesday, February 6, 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) .

   Find an equation of the form a x + b y + c z = d satisfied by all three of these points in two ways:

   1. Finding a normal vector, taking that as the coefficient vector (a, b, c), and then obtaining d by evaluating a x + b y + c z at any of the three given points.

   2. Assuming d = 1 (after dividing the equation by d) and then solving the three equations in a, b, c obtained from substituting each of the three given points into the general equation a x + b y + c z = 1.

   3. Are there cases when the second method will not work?

2. Let P be the point (3, -1, 2), and let V and W be the vectors V = (1, 3, -4) and W = (-2, 1, -1). Let F be the point-valued function of two variables given by

   F(s, t) = P + s V + t W .

   The set of points F(s, t) as (s, t) varies in R^{2} is a plane Pi in R^{3}.

   1. Find a vector U that is normal to Pi.

   2. Find a linear equation for Pi.

   3. Find a parametric representation for the line through the point P that is normal to the plane Pi.

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