Transcript Three-Dimensional Cartesian Coordinate System

Let’s start with a little problem…
Use the fact that k = 2p is twice the focal length and half the
focal width to determine a Cartesian equation of the parabola
whose polar equation is given.
12
4
r

3  3cos  1  cos 
 e  1, k  4
The graph???
k  2 p, so… p  2, 4 p  8
Vertex:
 h, k    2,0
And since the parabola opens left, the equation is:
y  8  x  2
2
Three-Dimensional
Cartesian
Coordinate System
Section 8.6a
Drawing
Practice:
(x, 0, z)
z
z = constant
(0, 0, z)
(0, y, z)
P(x, y, z)
(0, y, 0)
y
y = constant
(x, 0, 0)
(x, y, 0)
x
x = constant
Important Features of the 3-D Cartesian Coordinate System
Coordinate Axes – the axes labeled x, y, and z – they form the
right-handed coordinate frame.
Cartesian Coordinates of P – the real numbers x, y, and z that
make up an ordered triple (x, y, z), and locate point P in space.
Coordinate Planes – the xy-plane, the xz-plane, and the
yz-plane have equations z = 0, y = 0, and x = 0, respectively.
Origin – the point (0, 0, 0) where the coordinate planes meet.
Octants – the eight regions defined by the coordinate planes.
The first octant contains all points in space with three positive
coordinates.
Guided Practice
Draw a sketch that shows each of the following points.
 2,3,5
 4, 2,1
 3,6, 5
Equation of a Sphere
First, remind me of the definition of a circle:
Circle: the set of all points in a plane that lie a fixed distance
from a fixed point.
And the definition of a sphere?
Sphere: the set of all points that lie a fixed distance from a
fixed point.
fixed distance = radius fixed point = center
Now, do you recall the standard equation of a circle???
2
2
2
 x  h
y k  r
Equation of a Sphere
A point P (x, y, z) is on a sphere with center (h, k, l )
and radius r if and only if
2
2
2
2
 x  h
 y  k  z l  r
Quick Example: Write the equation for the sphere with its center
at (–8, –2, 1) and radius 4 3.
 x  8   y  2    z  1
2
2
How do we graph this sphere???
2
 48
New Equations
But first, remind me…
Distance formula in the 2-D Cartesian Coordinate System?
d
 x1  x2    y1  y2 
2
2
Midpoint formula in the 2-D Cartesian Coordinate System?
 x1  x2 y1  y2 
M 
,

2 
 2
Distance Formula
(Cartesian Space)
The distance d(P, Q) between the points P(x1 , y 1 , z )
1
and Q(x 2 , y 2 , z 2 ) in space is
d  P, Q  
 x1  x2    y1  y2    z1  z2 
2
2
2
Midpoint Formula
(Cartesian Space)
The midpoint M of the line segment PQ with endpoints
P(x 1 , y1 , z1 ) and Q(x 2 , y 2 , z 2 ) is
 x1  x2 y1  y2 z1  z2 
M 
,
,

2
2 
 2
A Quick Example
Find the distance between the points P(–2, 3, 1)
and Q(4, –1, 5), and find the midpoint of the line
segment PQ.
d  P, Q   2 17
M  1,1,3
Can we verify these answers with a graph?
Planes and Other
Surfaces
We have already learned that every line in the Cartesian
plane can be written as a first-degree (linear) equation in two
variables; every line can be written as
Ax  By  C  0
How about every first-degree equation in three
variables???
They all represent planes in Cartesian space!!!
Planes and Other
Surfaces
Equation for a Plane in Cartesian Space
Every plane can be written as
Ax  By  Cz  D  0
where A, B, and C are not all zero. Conversely, every
first-degree equation in three variables represents a
plane in Cartesian space.
Guided Practice
Sketch the graph of
12 x  15 y  20 z  60
Because this is a first-degree equation, its graph is a plane!
Three points determine a plane  to find them:
Divide both sides by 60:
x y z
  1
5 4 3
It’s now easy to see that the following points are on the plane:
5,0,0 0,4,0 0,0,3
Now where’s the graph???
Guided Practice
Sketch a graph of the given equation. Label all intercepts.
2y  z  6
x3