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 ax + by + cz = d satisfied by all three of these points in two ways:

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

(b) 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 ax + by + cz = 1.

(c) 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 + sV + tW .

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

(a) Find a vector U that is normal to Π.

(b) Find a linear equation for Π.

(c) Find a parametric representation for the line through the point P that is normal to the plane Π.

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