13-5 Coordinates in Space

By Danny Nguyen and Jimmy Nguyen

Objectives

Graph solids in space.Use the Distance and Midpoint Formulas for points in space.

Ordered Triples

In the coordinate plane we used an ordered pair with 2 real numbers to determine a point (x,y)

In space, we need 3 real numbers to graph a point. This is because space has 3 dimensions. These numbers make up an ordered triple (x,y,z).

Space In space, the x-, y-,

and z- axes are perpendicular to each other.

X represents the depth

Y represents the width Z represents the

height Notice how P(2,3,6) is

graphed. +

_

+

+_

_

Example 1: How to Graph a Shape in Space Graph a rectangular solid that

contains point A(-4,2,4) and the origin as vertices.

Example 1:How to Graph a Shape in Space Plot the x-coordinate

first. Go 4 units in the negative direction.

Next, plot the y-coordinate. Go 2 units in the positive direction.

Finally, plot the z- coordinate. 4 units in the positive direction

We have now plotted coordinate A.

Draw the rest of the rectangular prism.

Distance Formula in Space Remember Distance Formula from

the coordinate plane? We also have a formula for distance in Space.

Proof of the Distance Formula in Space

Example 2: Distance Formula

Find the Distance between T(6, 0, 0) and Q(-2, 4, 2).

Your Turn : (Distance Formula) Find the distance between A(3, 1, 4) and B(8, 2, 5)

AB

AB

( ) + ( ) + ( )( ) + ( ) + ( ) OR 3 3Answer:

√27

Midpoint Formula in Space We also have a formula for Midpoints

in Space.

Midpoint Formula Explanation An average is defined as the middle

measure of a data set. When we use midpoint formula, we

are basically finding the average between the x, y, and z, coordinates.

Putting the averages together to make an ordered triple lets us find where the midpoint of the segment is in space.

Example 3: Midpoint Formula Determine the coordinates of the midpoint M of . T(6, 0, 0) and Q(-2, 4, 2)

Your Turn: (Midpoint Formula) Find the coordinates of the midpoint M of

AB. A(3, 1, 4) and B(8, 2, 5)

= ( , , )Answer: (Secant), just kidding :P

it is (11/2, 3/2, 9/2) or (5.5, 1.5, 4.5)

Translating a Solid

Remember Translations? You can also do translations in space with solids.

It is basically the same principal we saw in Ch. 9 except we have another coordinate to translate.

Example 4: Translating a Solid Find the

coordinates of the vertices of the solid after the following translation. (x, y, z+20)

Example 4: Translating a Solid

Dilation with Matrices

We should also remember what a dilation is from Ch. 9. We used a matrix to find the coordinates of an image after a dilation. We can also do the same thing here.

Example 5: Dilation with Matrices Dilate the prism to

the right by a scale factor of 2. Graph the image after the dilation.

Example 5: Dilation with Matrices First, write a

vertex matrix for the rectangular prism.

Next, multiply each element of the vertex by the scale factor of 2.

Example 5: Dilation with Matrices We now have the

vertices of the dilated image.

To the right we have a graph of the dilated image.

Trololololololol… uhh Colby’s idea O.o Your homework:

Pre-AP Geometry: Pg 717 #11, 12, 14, 15-26, 28, 30

Have fun doing 16 problems!