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13-5 Coordinates in SpacePowerPoint Presentation

13-5 Coordinates in Space

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Objectives:

- Graph solids in space
- Use Distance & Midpoint Formulas for points in space

Vocab!

- 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

Graph a Rectangular Solid

B

A

z

- Graph a rectangular solid that has A(-5, 2, 4) and the origin as vertices.
- Plot the x-coordinate first. Draw a segment from the origin five units in 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.

F

E

C

D

x

O

G

y

Key Concepts

- Distance Formula in Space-
Given 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 Space-
Given 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

Example 1: Distance Formula in Space

- 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

Example 1: Midpoint Formula in Space

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)

Translating a Solid

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.

Dilation of Matrices

- Dilate the prism by a scale factor of 2.
- First, write a vertex matrix for the rectangular prism.
A B C D E F G H

x 0 0 3 3 3 3 0 0

y 0 2 2 0 0 2 2 0

z 0 0 0 0 1 1 1 1

- Next multiply each element of the vertex matrix by the scale factor, 2.
0 0 3 3 3 3 0 0 0 0 6 6 6 6 0 0

2 0 2 2 0 0 2 2 0 = 0 4 4 0 0 4 4 0

0 0 0 0 1 1 1 1 0 0 0 0 2 2 2 2

- First, write a vertex matrix for the rectangular prism.

The coordinates of the vertices of the dilated image are A’(0, 0, 0), B’(0, 4, 0), C’(6, 4, 0), D’(6, 0, 0), E’(6, 0, 2), F’(6, 4, 2), G’(0, 4, 2), H’(0, 0, 2).

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