11.1 Three-Dimensional Coordinate Systems

1. Draw the surface in .

http://matrix.skku.ac.kr/cal-lab/cal-11-1-1.html

Sol)

var('x,y,z');

implicit_plot3d(x+y==5, (x,-5,5),(y,-5,5),(z,-5,5))


 


2. Draw the surface in .

 

http://matrix.skku.ac.kr/cal-lab/cal-11-1-2.html

 Sol)

 

var('x,y,z')

A = plot3d(x^2-y^2==3,(x,-10,10),(y,-10,10), frame=True, color='blue', opacity=0.5)

A = A+line3d([(-10,0,0), (10,0,0)], arrow_head=True,rgbcolor='black')

A = A+text3d("x", (10,0.3,0), color='black')

A = A+line3d([(0,-10,0.3), (0,10,0)], arrow_head=True,rgbcolor='black')

A = A+text3d("y", (0,10,0.3), color='black')

A = A+line3d([(0,0,-25), (0,0,15)], arrow_head=True,rgbcolor='black')

A = A+text3d("z", (0,0.3,15), color='black')

show(A)

 

 

3. Find the lengths of the sides of the triangle with vertices , and , Is a right triangle? Is it an isosceles triangle?

http://matrix.skku.ac.kr/cal-lab/cal-11-1-3.html

 Sol) Isosceles triangle

 

A=(1,3,-2)

B=(3,1,-3)

C=(2,-1,-1)

show(point3d([p1,p2,p3])+line([p1,p2])+line([p2,p3])+line([p3,p1]))

AB=sqrt((1-3)^2+(3-1)^2+(-2-(-3))^2);

BC=sqrt((3-2)^2+(1-(-1))^2+(-3-(-1))^2);

CA=sqrt((2-1)^2+(-1-3)^2+(-1-(-2))^2);

print AB

print BC

print CA

  3

  3

  3*sqrt(2)

 

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

http://matrix.skku.ac.kr/cal-lab/cal-11-1-4.html

 Sol)

 

a=sqrt(x[0]^2)

b=sqrt(x[1]^2)

c=sqrt(x[2]^2)

d=sqrt(x[1]^2 +x[2]^2)

e=sqrt(x[0]^2 +x[2]^2)

f=sqrt(x[0]^2 +x[1]^2)

a,b,c,d,e,f

  (2, 6, 3, 3*sqrt(5), sqrt(13), 2*sqrt(10))


5. Find an equation of the sphere with center and radius 3. What is the intersection of this sphere with the -plane?

 Sol) An enquation of the sphere : , and

     the intersection of this sphere with the -plane can be obtained by in the equation. Hence 

     .


6. Find an equation of the sphere that passes through the point and has center .

 Sol) 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.

 Sol)

      . Hence

         center: , radius: .


8.

 Sol) 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. .

 Sol) The equation represents a plane parallel to the -plane and 8 units in front of it. So the answer is a plane parallel to the -plane and 8 units in front of it.


11. .


12. .

 Sol) 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 .


13. , .

http://matrix.skku.ac.kr/cal-lab/cal-11-1-13.html

 Sol)

 

var('x,y,z');

implicit_plot3d(x^2+y^2==3, (x,-5,5),(y,-5,5),(z,-1,-1))


 


14. .

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


15. .

 Sol) 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 .


16. .

http://matrix.skku.ac.kr/cal-lab/cal-11-1-16.html

 Sol)

 

x, y, z = var('x, y, z')

implicit_plot3d(x^2 + z^2 == 9 -2*z, (x,-5, 5), (y,-5, 5), (z,-5, 5), opacity=0.2, color="red")

 


17-18. Describe the given region by an inequality.


17. The half-space consisting of all points to the left of the -plane.

 Sol) This describes all points with positive -coordinates, that is, .

 

18. The solid rectangular box in the first octant bounded by the planes , , and .

http://matrix.skku.ac.kr/cal-lab/cal-11-1-18.html

 Sol)

 

x, y, z = var('x, y, z')

p1=implicit_plot3d(x == 1, (x,-5, 5), (y,-5, 5), (z,-5, 5), opacity=0.2, color="red")

p2=implicit_plot3d(y == 3, (x,-5, 5), (y,-5, 5), (z,-5, 5), opacity=0.2, color="blue")

p3=implicit_plot3d(z == 2, (x,-5, 5), (y,-5, 5), (z,-5, 5), opacity=0.2, color="green")

show(p1+p2+p3)