11.1 Three-Dimensional Coordinate Systems by SGLee-HSKim -JHLee
1. Draw the surface in .
(You may do it with Sage in http://math1.skku.ac.kr/
Refer open resources in http://math1.skku.ac.kr/pub/ )
2. Draw the surface in .
3. Find the lengths of the sides of the triangle with vertices , and .
Is a right triangle? Is it an isosceles triangle?
Isosceles triangle because AB=BC.
4. Find the distance from to each of the following.
(a) The -axis (b) The -axis (c) The -axis
(d) The -plane (e) The -plane (f) The -plane
5. Find an equation of the sphere with center and radius 3.
What is the intersection of this sphere with the -plane?
An equation of the sphere : ,
and the intersection of this sphere with the -plane can be obtained by substituting in the equation.
6. Find an equation of the sphere that passes through the point and has center .
The distance between and is the radius of the sphere.
Thus, an equation of sphere is .
7-8. Show that the equation represents a sphere, and find its center and radius.
center: , radius: .
Completing squares in the equation gives :
, with the center and radius .
9. (a) Prove that the midpoint of the line segment from
to is , .
(b) Find the lengths of the medians of the triangle with vertices , and .
10-16. Determine the region of represented by the equation or inequality.
The equation represents a plane parallel to the -plane and 8 units in front of it.
The inequality represents all points on or between the horizontal planes (the -plane) and .
So the answer is all points on or between the horizontal plane (the -plane) and .
The set of all points in whose distance from the -axis is .
This is a cylinder of radius 3 and axis along -axis.
The inequality is equivalent to .
So the region consists of those points whose distance from the point is greater than 1.
This is the set of all points outside the sphere with radius 1 and center .
17-18. Describe the given region by an inequality.
17. The half-space consisting of all points to the left of the -plane.
This describes all points with positive -coordinates, that is, .
18. The solid rectangular box in the first octant bounded by the planes , , and .