Calculus-Sec-11-1-Solution

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.

Hence .

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.

Hence, .

Thus, an equation of sphere is .

7-8. Show that the equation represents a sphere, and find its center and radius.

7.

. Hence

8.

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.

10.

The equation  represents a plane parallel to the -plane and 8 units in front of it.

11.

12.

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 .

14.

The set of all points in  whose distance from the -axis is .

This is a cylinder of radius 3 and axis along -axis.

15.

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 .