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Exploring 3D Coordinates: Graphing, Transformations, and Distance in Space

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This guide focuses on understanding the representation of points in three-dimensional space through coordinates. It covers the fundamentals of graphing rectangular solids, using distance and midpoint formulas, and the concepts of translating and dilating solids. Each point requires three coordinates (x, y, z) that correspond to depth, width, and height. Examples illustrate how to graph a rectangular prism, find distances and midpoints between points, and apply transformations to solid shapes. Ideal for students learning geometry in higher dimensions.

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Exploring 3D Coordinates: Graphing, Transformations, and Distance in Space

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  1. 13.5 Coordinates in Space By: EmilySchneider Lindsey Grisham

  2. Mission • Graph a rectangular solid • Use the Distance point and Midpoint Formulas in space. • Translating solids • Dilating solids

  3. 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 y x

  4. Facts about Space • X- represents the depth • Y- represents the width • Z- represents the height

  5. 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 y x

  6. Example 1 • Graph a rectangular solid that contains point A(-4,2,4) and the origin as vertices.

  7. Example 1 z y x

  8. Example 1 ~ Answer

  9. Formulas Distance formula for space: _____________________________________ Midpoint Formula for space:

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

  11. Example 2~ Answer Distance= =√[6-(-2) 2 + (o-4) 2 + (0-2) 2 = √(64+ 16 + 4) Answer= √84 or 2√21

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

  13. Example 3~ Answer M of = = = (2, 2, 1)

  14. 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.

  15. Example 4 Find the coordinates of the vertices of the solid after the following translation. (x, y, z+20)

  16. Example 4~ answer

  17. 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.

  18. Example 5 • Dilate the prism to the left by a scale factor of 2. Graph the image after the dilation.

  19. 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

  20. Example 5 ~ answer Original coordinates Translated coordinates Scale factor

  21. Example 5 • Now, we have the vertices of the dilated image. • The right is the dilated image graphed.

  22. Assignment Page 717 #10-15, 16-20 evens, 23-26, 35

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