2. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes.
3. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes. There are two ways to add the zâaxis.
4. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes. There are two ways to add the zâaxis.
x
y
z+
Rightâhand systemLeftâhand system
x
y
z+
5. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes.
x
y
z+
Rightâhand systemLeftâhand system
x
y
z+
Rightâhand systemLeftâhand system
There are two ways to add the zâaxis.
6. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes.
x
y
z+
Rightâhand systemLeftâhand system
x
y
z+
Rightâhand systemLeftâhand system
In math/sci, we use the
rightâhand system.
The left hand system is
used in computer
graphics for the virtual
space beyond the screen.
There are two ways to add the zâaxis.
7. 3D Coordinate System
To set up the 3D coordinate system, we add
a zâaxis which is perpendicular to both the
x&y axes.
x
y
z+
Rightâhand systemLeftâhand system
x
y
z+
Rightâhand systemLeftâhand system
In math/sci, we use the
rightâhand system.
The left hand system is
used in computer
graphics for the virtual
space beyond the screen.
We write the 2D plane
and the 3D space
respectively as R2 and R3.
There are two ways to add the zâaxis.
8. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
9. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:
10. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
11. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
x
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
12. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
x
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(â2, â1, â3)
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
13. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(â2, â1, â3)
x
A(2, 0 , 0),
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
14. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(â2, â1, â3)
x
A(2, 0 , 0),
(1, 3, 0),
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
15. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(â2, â1, â3)
x
A(2, 0 , 0),
B(1, 3, 4),
(1, 3, 0),
16. 3D Coordinate System
Every position in space, may be addressed by
three numbers (x, y, z), called an ordered triple.
y
z+
Given (x, y, z), to find the
location it represents:
1. find (x, y) in the x&y
coordinate plane
2. the z gives the location
of the point, z units above
or below (x, y).
z > 0 ď above
z < 0 ď below
Ex: Draw A(2, 0 , 0), B(1, 3, 4), C(â2, â1, â3)
x
A(2, 0 , 0),
B(1, 3, 4),
C(â2, â1, â3)
(1, 3, 0),
(â2, â1, 0),
18. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
Eye:(1, â1, 1)
19. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1)
20. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1) Eye:(1, â1, â1)
21. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1) Eye:(1, â1, â1)
There are three coordinate planes:
y
z+
x
22. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
y
z+
xyâplane
x
y
z+
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1) Eye:(1, â1, â1)
There are three coordinate planes:
the xyâplane,
x
23. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1) Eye:(1, â1, â1)
y
z+
xyâplane
xzâplane
There are three coordinate planes:
the xyâplane,
the xzâplane,
x
24. 3D Coordinate System
3D coordinate may be drawn from different eye
positions:
x
y
z+
x
y
z+
x
y
z+
Eye:(1, â1, 1) Eye:(1, 1, 1) Eye:(1, â1, â1)
There are three coordinate planes:
the xyâplane,
the xzâplane,
and the yzâplane.
x
y
z+
xyâplane
xzâplane
yzâplane
Points in the xy plane are
(x, y, 0), i.e. defined by z = 0.
26. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
27. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
28. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yzâplane
y
29. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yzâplane x = 4
y
30. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
x
z+
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
Example: a. x = 4 is
a plane // to the yzâplane x = 4
b. y = 4 is a plane //
to the xzâplane
y
31. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yzâplane x = 4
b. y = 4 is a plane //
to the xzâplane
yy = 4
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
32. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yzâplane x = 4
b. y = 4 is a plane //
to the xzâplane
yy = 4
c. z = 4 is a plane //
to the xyâplane
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
33. 3D Coordinate System
x
z+
Example: a. x = 4 is
a plane // to the yzâplane x = 4
b. y = 4 is a plane //
to the xzâplane
yy = 4
c. z = 4 is a plane //
to the xyâplane
z = 4
Some Basic Equations and Their Graphs in 3D
In general, the graph of an equation in R3 is a surface.
The constant equations:
The graphs of the equations
x=k, y=k, or z=k are planes
that are parallel to the coordinate
planes.
35. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Some Basic Equations and Their Graphs in 3D
36. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Some Basic Equations and Their Graphs in 3D
37. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Some Basic Equations and Their Graphs in 3D
38. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, â4, 0)
39. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Plot the y and z intercepts, the line
â y + 2z = 4 in the yzâplane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, â4, 0)
40. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Since the equation doesn't have x, so x may assume any
value.
Plot the y and z intercepts, the line
â y + 2z = 4 in the yzâplane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, â4, 0)
41. 3D Coordinate System
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line ây + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the xâaxis.
Plot the y and z intercepts, the line
â y + 2z = 4 in the yzâplane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
x
z+
y
(0, 0, 2)
(0, â4, 0)
42. 3D Coordinate System
x
z+
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
y
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line ây + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the xâaxis.
All such parallel lines passing through ây + 2x = 4 form a
plane, that is parallel to the xâaxis, is the graph.
Plot the y and z intercepts, the line
â y + 2z = 4 in the yzâplane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
(0, 0, 2)
(0, â4, 0)
43. 3D Coordinate System
x
z+
Example: Sketch â y + 2z = 4 in 3D
Set x=y=0 ď z=2, get (0, 0 ,2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0, it not possible, so
there is no x intercept.
y
Since the equation doesn't have x, so x may assume any
value. Hence a point (0, a, b) on the line ây + 2z = 4 gives
infinite many solutions (#, a, b) and they form a line parallel to
the xâaxis.
All such parallel lines passing through ây + 2x = 4 form a
plane, that is parallel to the xâaxis, is the graph.
Plot the y and z intercepts, the line
â y + 2z = 4 in the yzâplane is part
of the graph.
Some Basic Equations and Their Graphs in 3D
(0, 0, 2)
(0, â4, 0)
44. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes.
Some Basic Equations and Their Graphs in 3D
45. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Some Basic Equations and Their Graphs in 3D
46. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Some Basic Equations and Their Graphs in 3D
47. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Some Basic Equations and Their Graphs in 3D
48. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Some Basic Equations and Their Graphs in 3D
49. 3D Coordinate System
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0 ď x=2, get (2, 0, 0).
Three points determine a plane.
Some Basic Equations and Their Graphs in 3D
50. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0 ď x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D
51. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0 ď x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D
(2, 0, 0)
(0, â4, 0)
(0, 0, 2)
52. 3D Coordinate System
x
z+
The linear equations:
The graphs of ax + by + cz = d are
planes. Use the intercepts to graph:
Set x=y=0 to get the z intercept,
set x=z=0 to get the y intercept,
set y=z=0 to get the x intercept.
Example: Sketch 2x â y + 2z = 4
Set x=y=0 ď z=2, get (0, 0, 2)
Set x=z=0 ď y=â4, get (0, â4, 0)
Set y=z=0 ď x=2, get (2, 0, 0).
y
Three points determine a plane. Plot these three
intercepts and the plane containing them is the graph.
Some Basic Equations and Their Graphs in 3D
(2, 0, 0)
(0, â4, 0)
(0, 0, 2)
53. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
54. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Example: Sketch z = x2
55. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Example: Sketch z = x2
Draw z = x2 in the xzâplane which is a parabola.
56. 3D Coordinate System
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
Example: Sketch z = x2
Draw z = x2 in the xzâplane which is a parabola.
y
z
x
57. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
Example: Sketch z = x2
Draw z = x2 in the xzâplane which is a parabola. Slide this
parabola in the y (the missing variable) direction we get the
(parabolic) cylinderâsurface as shown.
58. 3D Coordinate System
y
z
x
Some Basic Equations and Their Graphs in 3D
General Cylinders (equation with a missing variable)
If an equation doesn't contain a particular
variable, that variable may take on any
value. To draw its graph, draw the 2D graph
in the coordinate plane of the variables in
the equation.
Slide the 2D graph parallelwise
in the direction of the missing variable,
the surface it forms is the 3D graph.
This is called a general cylinder.
Example: Sketch z = x2
Draw z = x2 in the xzâplane which is a parabola. Slide this
parabola in the y (the missing variable) direction we get the
(parabolic) cylinderâsurface as shown.
59. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D = ď Îx2 + Îy2 + Îz2
= ď(x2 â x1)2 + (y2 â y1)2 + (z2 â z1)2
60. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D = ď Îx2 + Îy2 + Îz2
= ď(x2 â x1)2 + (y2 â y1)2 + (z2 â z1)2
Example: The distance between (2, â1, 1) and (1, â1, 3) is
Îx = 1, Îy = 0, Îz = â2, so D = ď1+ 0 + 4 = ď5
61. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D = ď Îx2 + Îy2 + Îz2
= ď(x2 â x1)2 + (y2 â y1)2 + (z2 â z1)2
Example: The distance between (2, â1, 1) and (1, â1, 3) is
Îx = 1, Îy = 0, Îz = â2, so D = ď1+ 0 + 4 = ď5
MidâPoint Formula: The midâpoint in 3D is computed
coordinateâwise so the midâpoint of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )
62. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D = ď Îx2 + Îy2 + Îz2
= ď(x2 â x1)2 + (y2 â y1)2 + (z2 â z1)2
Example: The distance between (2, â1, 1) and (1, â1, 3) is
Îx = 1, Îy = 0, Îz = â2, so D = ď1+ 0 + 4 = ď5
MidâPoint Formula: The midâpoint in 3D is computed
coordinateâwise so the midâpoint of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )
y
z+
xA(2, 0 , 0)
B(1, 3, 4)
63. 3D Coordinate System
Distance Formula in 3D:
The distance between (x1, y1, z1), (x2, y2, z2) is:
D = ď Îx2 + Îy2 + Îz2
= ď(x2 â x1)2 + (y2 â y1)2 + (z2 â z1)2
Example: The distance between (2, â1, 1) and (1, â1, 3) is
Îx = 1, Îy = 0, Îz = â2, so D = ď1+ 0 + 4 = ď5
MidâPoint Formula: The midâpoint in 3D is computed
coordinateâwise so the midâpoint of
(x1, y1, z1), (x2, y2, z2) is:
(
x1+ x2 y1+ y2 z1+ z2
2 2 2, , )
y
z+
xA(2, 0 , 0)
B(1, 3, 4)
midâpt
(3/2, 3/2, 2)
(3/2, 3/2, 0)
65. 3D Coordinate System
Equations of Spheres
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x â a)2 + (y â b)2 + (z â c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
66. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x â a)2 + (y â b)2 + (z â c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(xâa)2+(yâb)2+(zâc)2=r2
67. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x â a)2 + (y â b)2 + (z â c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(xâa)2+(yâb)2+(zâc)2=r2
x2+y2+z2=r2
Equations of Ellipsoid
The graph of the equation
is the ellipsoid centered at (a, b ,c),
with
xâradius=r, yâradius=s, zâradius=t
(x â a)2 (y â b)2 (z â c)2
r2 s2 t2 = 1++
68. 3D Coordinate System
x
z+
Equations of Spheres
y
Some Basic Equations and Their Graphs in 3D
The equation of the sphere with radius r,
centered at (a, b ,c) is
(x â a)2 + (y â b)2 + (z â c)2 = r2.
In particular x2 + y2 + z2 = r2 is the
sphere centered at (0, 0, 0) with radius r.
(a, b, c)
r
r
(xâa)2+(yâb)2+(zâc)2=r2
x2+y2+z2=r2
Equations of Ellipsoid
The graph of the equation
is the ellipsoid centered at (a, b ,c),
with
xâradius=r, yâradius=s, zâradius=t
(x â a)2 (y â b)2 (z â c)2
r2 s2 t2 = 1++
x
z+
y
(a, b, c)
r
s
t
69. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, â5) as a diameter.
b. What is the highest point on this sphere?
70. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, â5) as a diameter.
The center of the sphere is the midâpoint of the
two given points. Use the midâpoint formula,
we've the center = (3, 2, â1).
b. What is the highest point on this sphere?
71. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, â5) as a diameter.
The center of the sphere is the midâpoint of the
two given points. Use the midâpoint formula,
we've the center = (3, 2, â1). The radius is half
of the length of the diameter, so using the
distance formula r = ½ď22+22+82 = 3ď2 and the
equation is (x â 3)2 + (y â 2)2 + (z + 1)2 = 18.
b. What is the highest point on this sphere?
72. 3D Coordinate System
Example: a. Find the equation of the sphere
which has (2, 1, 3), (4, 3, â5) as a diameter.
The center of the sphere is the midâpoint of the
two given points. Use the midâpoint formula,
we've the center = (3, 2, â1). The radius is half
of the length of the diameter, so using the
distance formula r = ½ď22+22+82 = 3ď2 and the
equation is (x â 3)2 + (y â 2)2 + (z + 1)2 = 18.
b. What is the highest point on this sphere?
The highest point is 3ď2 above the center
so itâs (3, 2, â1+3ď2).