13.5 Coordinates in Space

By: Emily SchneiderLindsey Grisham

Mission Graph a rectangular solid Use the Distance point and Midpoint

Formulas in space. Translating solids Dilating solids

Graphing In space, each

point requires three coordinates. This is because space has three dimensions.

The x-, y-, and z-axes are all perpendicular to each other.

A point in space is represented by an ordered triple.

z

yx

Facts about Space X- represents the depth

Y- represents the width

Z- represents the height

Graphing a Rectangular Prism

Plot the x-coordinate first. Draw a segment from the origin _ units in the ± direction.

To plot the y-coordinate, draw a segment _ units in the ± direction.

Next, to plot the z-coordinate draw a segment _ units in the ± direction.

Label the point Draw a rectangular prism

and label each vertex.

z

yx

Example 1 Graph a rectangular solid that

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

Example 1

z

y

x

Example 1 ~ Answer

FormulasDistance formula for space:

_____________________________________

Midpoint Formula for space:

Example 2 (Distance)

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

Example 2~ AnswerDistance=

= √[6-(-2) 2 + (o-4) 2 + (0-2) 2

= √(64+ 16 + 4)

Answer= √84 or 2√21

Example 3(Midpoint)

Determine the coordinates of the midpoint M of T(6, 0, 0) and Q(-2, 4, 2)

Example 3~ Answer∞ M of = =

= (2, 2, 1)

Translations¤ In chapter 9 we learned how to translate a 2

dimensional shape.

¤ The same concept applies for translating a 3 dimensional shape.

¤ However, we have another coordinate (z) that we need to translate.

¤ First, write all of the vertices of the preimage in a chart.

¤ Next, add the ‘scale factor’ to the axis it specifies.

Example 4Find the coordinates of the

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

Example 4~ answer

Dilation using Matrices In chapter 9 we used a matrix to find the

coordinates of a dilated image.

The same concept works in space.

First, write a matrix for the vertexes of the rectangular prism.

Then, multiply the whole matrix by the scale factor.

Example 5 Dilate the prism

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

Example 5∫ First, write a matrix

for the vertexes of the rectangular prism.

∫ Then, multiply the whole matrix by the scale factor.

∫ Dilate these coordinates with a scale factor of 2.

Original coordinates

Example 5 ~ answer

Original coordinates

Translated coordinates

Scale factor

Example 5 Now, we have the

vertices of the dilated image.

The right is the dilated image graphed.

Assignment

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