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(b) Find the two planes described in part (a) corresponding to the triplets of points in Exercises 4(a) and 4(b).
Find the equations of the spheres in 3-space that pass through the following points:
6.
(a) (1, 2, 3), (−1, 2, 1), (1, 0, 1), (1, 2, −1)
(b) (0, 1, −2), (1, 3, 1), (2, −1, 0), (3, 1, −1)
Show that Equation 10 is the equation of the conic section that passes through five given distinct points in the plane.
7.
Show that Equation 11 is the equation of the plane in 3-space that passes through three given noncollinear points.
8.
Show that Equation 12 is the equation of the sphere in 3-space that passes through four given noncoplanar points.
9.
Find a determinant equation for the parabola of the form
10.
that passes through three given noncollinear points in the plane.
What does Equation 9 become if the three distinct points are collinear?
11.
What does Equation 11 become if the three distinct points are collinear?
12.
What does Equation 12 become if the four points are coplanar?
13.
Section 11.1
Technology Exercises
The following exercises are designed to be solved using a technology utility. Typically, this will be MATLAB, Mathematica, Maple,
Derive, or Mathcad, but it may also be some other type of linear algebra software or a scientific calculator with some linear algebra
capabilities. For each exercise you will need to read the relevant documentation for the particular utility you are using. The goal of
these exercises is to provide you with a basic proficiency with your technology utility. Once you have mastered the techniques in
these exercises, you will be able to use your technology utility to solve many of the problems in the regular exercise sets.
The general equation of a quadric surface is given by
T1.
Given nine points on this surface, it may be possible to determine its equation.
