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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