Download 13-5 Coordinates in Space

By: Robert Flipping and Blake Munoz
Fourth hour
 Graph
solids in space
 Use Distance & Midpoint
Formulas for points in space


Ordered triple- A point in space is represented
by an ordered triple of real numbers (x, y, z)
Each vertex is labeled with its corresponding “x” “y” and “z”
value
z


Graph a rectangular solid that has
x
A(-5, 2, 4) and the origin as vertices.
F
E
C
D
 Plot the x-coordinate first. Draw a
segment from the origin five units in
O
G
y
the negative direction.
 Plot the y-coordinate two units in
the positive direction
Next, to plot the z-coordinate, draw a segment four units
long in the positive direction.
Draw the rectangular prism and label each vertex.
B
A



Distance Formula in SpaceGiven two points A(x1, y1, z1) and B(x2, y2, z2)
in space, the distance between A and B is given
by the following equation:
√(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
Midpoint Formula in SpaceGiven two points A(x1, y1, z1) and B(x2, y2, z2)
in space, the midpoint of segment AB is at:
x1+x2 , y1+y2 , z1+z2
2
2
2
Determine the distance between T(6, 0, 0) and
Q(-2, 4, 2).
TQ = √(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
TQ = √[6 – (-2)]2 + (0 - 4)2 + (0 - 2)2
TQ = √84 or 2√21


T(6, 0, 0)
Q(-2, 4, 2)
Determine the coordinates of the midpoint M
of segment TQ.
M= x1+x2 , y1+y2 , z1+z2
2
2
2
M= 6 + (-2) , 0 + 4 , 0 + 2
2
2
2
M = (2, 2, 1)
Suppose an elevator is 5 feet wide, 6 feet
deep, and 8 feet tall. Position the
elevator on the ground floor at the
origin of a three dimensional space.
If the distance between the floors of a
warehouse is 10 feet, write the
coordinates of the vertices of the
elevator after going up to the third
floor.
Since the elevator is a rectangular prism,
use positive values for x, y, and z.
Write the coordinates of the corner.
The points on the elevator will rise 10
feet for each floor. When the elevator
ascends to the third floor it will have
traveled 20 feet.
Use the translation (x, y, z)  (x, y, z +
20) to find the coordinates of each
vertex of the rectangular prism that
represents the elevator.
Coordinates of the
vertices, (x, y, z)
Preimage
Translated
coordinates,
(x, y, z + 20)
Image
J(0, 5, 8)
J’(0, 5, 28)
K(6, 5, 8)
K’(6, 5, 28)
L(6, 0, 8)
L’(6, 0, 28)
M(0, 0, 8)
M’(0, 0, 28)
N(6, 0, 0)
N’(6, 0, 20)
O(0, 0, 0)
O’(0, 0, 20)
P(0, 5, 0)
P’(0, 5, 20)
Q(6, 5, 0)
Q’(6, 5, 20)

Dilate the prism by a scale factor of 2.

First, write a vertex matrix for the rectangular prism.
ABCDEFGH

x 00333300
y 02200220
z 00001111
Next multiply each element of the vertex matrix by
The coordinates of
the scale factor, 2.
the vertices of the
00333300
00666600
dilated image are
A’(0, 0, 0), B’(0, 4, 0),
2 02200220 = 04400440
C’(6, 4, 0), D’(6, 0, 0),
00001111
00002222
E’(6, 0, 2), F’(6, 4, 2),
G’(0, 4, 2), H’(0, 0, 2).