13.5 Coordinates in Space
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Transcript of 13.5 Coordinates in Space
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13.5 Coordinates in Space
By: Emily SchneiderLindsey Grisham
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Mission Graph a rectangular solid Use the Distance point and Midpoint
Formulas in space. Translating solids Dilating solids
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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
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Facts about Space X- represents the depth
Y- represents the width
Z- represents the height
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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
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Example 1 Graph a rectangular solid that
contains point A(-4,2,4) and the origin as vertices.
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Example 1
z
y
x
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Example 1 ~ Answer
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FormulasDistance formula for space:
_____________________________________
Midpoint Formula for space:
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Example 2 (Distance)
* Find the Distance between T(6, 0, 0) and Q(-2, 4, 2).
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Example 2~ AnswerDistance=
= √[6-(-2) 2 + (o-4) 2 + (0-2) 2
= √(64+ 16 + 4)
Answer= √84 or 2√21
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Example 3(Midpoint)
Determine the coordinates of the midpoint M of T(6, 0, 0) and Q(-2, 4, 2)
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Example 3~ Answer∞ M of = =
= (2, 2, 1)
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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.
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Example 4Find the coordinates of the
vertices of the solid after the following translation. (x, y, z+20)
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Example 4~ answer
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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.
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Example 5 Dilate the prism
to the left by a scale factor of 2. Graph the image after the dilation.
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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
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Example 5 ~ answer
Original coordinates
Translated coordinates
Scale factor
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Example 5 Now, we have the
vertices of the dilated image.
The right is the dilated image graphed.
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Assignment
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#10-15, 16-20 evens,23-26, 35