11.1     Three-Dimensional Coordinate Systems     by SGLee-HSKim -JHLee



 1. Draw the surface  in .




     (You may do it with Sage in

      Refer open resources in )

 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.






       . Hence

          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 .






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