California State University, Northridge - Northridge, CA
Course Name: Calculus III (Math 250)
Professor Name: Dr. Michael Johnson
Assignment Title: 3D Coordinate System Homework Exercises
Submission Date: August 01, 2025
Math 250 3D Coordinate System Homework Exercises
Exercises
1. Point Projection Problem
Question: In a 3D coordinate system, what are the projection points of point P(4, -2,
5) onto the xy-plane, yz-plane, and xz-plane respectively?
Solution:
• The projection onto the xy-plane has z-coordinate 0: (4, -2, 0)
• The projection onto the yz-plane has x-coordinate 0: (0, -2, 5)
• The projection onto the xz-plane has y-coordinate 0: (4, 0, 5)
Answer: Projection onto xy-plane: (4, -2, 0); onto yz-plane: (0, -2, 5); onto xz-plane:
(4, 0, 5)
2. Distance Calculation Between Two Points
Question: Calculate the distance between points A(1, 2, 3) and B(4, 6, 8).
Solution:
Using the distance formula in 3D space:
d(A , B)=
√
¿¿
Substituting the coordinates:
d(A , B)=
√
¿¿
Answer:
5
√
2
(or approximately 7.07)
3. Sphere Equation Construction
Question: Write the equation of a sphere with center C(2, -3, 4) and radius r=5.
Solution:
The standard equation of a sphere is:
¿
where (h, k, l) is the center and r is the radius. Substituting the given values:
¿
Answer:
¿
4. Sphere Equation Identification
Question: The equation
x2+y2+z2−6x+4y − 2z− 11=0
represents a sphere. Find its
center coordinates and radius.
Solution:
We need to complete the square for x, y, and z:
x2−6x+y2+4y+z2−2z=11
(x2−6x+9)+( y2+4y+4)+( z2−2z+1)=11 +9+4+1
¿
Thus, the center coordinates are (3, -2, 1) and the radius r=5.
Answer: Center (3, -2, 1), radius 5
5. Equation Graph Identification
Question: What does the equation
x2+z2=4
represent in the R³ coordinate system?
Solution:
In R³, the equation
x2+z2=4
doesnt contain the y-variable, meaning it holds for all
values of y. This is a cylindrical surface with radius 2 and central axis along the y-
axis.
Answer: A cylindrical surface with radius 2 and central axis along the y-axis
6. Coordinate Plane Equations
Question: Write the equations of the xy-plane, yz-plane, and xz-plane.
Solution:
• For the xy-plane, all points have z-coordinate 0: z=0
• For the yz-plane, all points have x-coordinate 0: x=0
• For the xz-plane, all points have y-coordinate 0: y=0
Answer: xy-plane: z=0; yz-plane: x=0; xz-plane: y=0
7. Distance from Point to Coordinate Planes
Question: Calculate the distance from point P(3, -4, 5) to each of the three
coordinate planes.
Solution:
The distance from a point to a coordinate plane equals the absolute value of the
corresponding coordinate:
• Distance to xy-plane (z=0): z = = 5
• Distance to yz-plane (x=0): x = = 3
• Distance to xz-plane (y=0): y = = 4
Answer: Distance to xy-plane: 5; to yz-plane: 3; to xz-plane: 4
8. Comprehensive Distance Problem
Question: Find the midpoint coordinates between points A(1, 2, 3) and B(4, 6, 8),
and calculate the distance from this midpoint to the origin.
Solution:
First, find the midpoint coordinates:
M
(
1+4
2,2+6
2,3+8
2
)
=M(2.5 ,4,5.5)
Then calculate the distance from the midpoint to the origin:
d=
√
¿¿
Answer: Midpoint coordinates (2.5, 4, 5.5), distance to origin
√
210
2
(or
approximately 7.245)
9. Sphere and Coordinate Plane Intersection
Question: For the sphere with equation
¿
, find the equation of the circle formed by
its intersection with the xy-plane.
Solution:
The equation of the xy-plane is z=0. Substitute into the sphere equation:
¿
¿
¿
This is a circle with center (2, 3, 0) and radius 3.
Answer:
¿
(at z=0)
10. Graph Comparison in Different Coordinate Systems
Question: What does the equation
y=4
represent in R, R², and R³ coordinate
systems respectively?
Solution:
• In R (1D), y=4 represents a single point
• In R² (2D), y=4 represents a horizontal line
• In R³ (3D), y=4 represents a plane parallel to the xz-plane
Answer: A point in R, a line in R², and a plane in R³
