Three-Dimensional Coordinates

Transcription

1 CHAPTER Three-Dimensional Coordinates Three-dimensional movies superimpose two slightl different images, letting viewers with polaried eeglasses perceive depth (the third dimension) on a two-dimensional screen. You will learn to work with points, lines, planes, and solid forms in a three-dimensional coordinate space.. Into the Third Dimension Point Plotting in Three Dimensions p. 89. Planes Equations of Planes p Distance Distance in Three Dimensions p.. Volume in Space Models and Volume p. Chapter l Three-Dimensional Coordinates 87

2 88 Chapter l Three-Dimensional Coordinates

3 . Into the Third Dimension Point Plotting in Three Dimensions Objectives In this lesson ou will: l Develop the three-dimensional coordinate sstem. l Graph points in three dimensions. Ke Terms l three-dimensional coordinate sstem l -plane l -plane l -plane l right-hand orientation l octant Problem Points, Lines, and Planes A point is a location in space that has no sie or shape, and is represented as a dot. A line is described as a straight continuous arrangement of an infinite number of points. Line segments and ras can be created from lines. A ra is a portion of a line that begins with a single point and etends infinitel in one direction. A line segment is a portion of a line that includes two points and all of the collinear points between the two points. A plane consists of an infinite number of lines that are placed side b side. Additional figures can be created within a plane using points, lines, line segments, and ras. These figures can be represented in the two-dimensional Cartesian coordinate plane. Areas, perimeters, and measures of two-dimensional figures can be calculated. Space consists of an infinite number of planes that are placed side b side. Additional figures can be created within space using points, lines, line segments, ras, planes, and two-dimensional figures. Volumes and surface areas of three-dimensional figures can be calculated. Lesson. l Point Plotting in Three Dimensions 89

4 . Draw a point.. Imagine that the point ou drew in Question is made of an infinitel elastic, stretchable material. Draw the figure that results if ou were to pull the point from the left and right in an infinite direction. What is the resulting figure? 3. A number line consists of an origin, indicated as the position of the number ero, and a distance designated as one unit. Draw a number line using the line that ou drew in Question.. Graph the points and on the number line ou drew in Question 3.. Imagine that the line ou drew in Question is made of an infinitel elastic, stretchable material. Draw the figure that results if ou were to pull the entire line from the top and bottom in an infinite direction. What is the resulting figure? 90 Chapter l Three-Dimensional Coordinates

5 6. Imagine that the number line ou drew in Question 3 is made of an infinitel elastic, stretchable material. Draw the figure that results if ou were to pull the entire number line from the top and bottom in an infinite direction. What is the resulting figure? 7. Graph the points (, 3) and (, 7) on the figure ou drew in Question Imagine that the figure ou drew in Question 6 is made of an infinitel elastic, stretchable material. Draw the figure that results if ou were to pull the entire figure from the front and back in an infinite direction. Lesson. l Point Plotting in Three Dimensions 9

6 A three-dimensional coordinate sstem is a sstem that includes -, -, and -aes drawn perpendicular to each other to represent points in space. 0 The -plane is a plane that contains the lines representing the - and -aes. The -plane is indicated b the lightest shade of gre. The -plane is a plane that contains the lines representing the - and -aes. The -plane is indicated b the middle shade of gre. The -plane is a plane that contains the lines representing the - and -aes. The -plane is indicated b the darkest shade of gre. Right-hand orientation is an orientation of the -, -, and -aes in which the positive -ais points in the direction of the first finger of the right hand, the positive -ais points in the direction of the second finger, and the positive -ais points in the direction of the thumb. 9 Chapter l Three-Dimensional Coordinates

7 The full three-dimensional coordinate sstem can be drawn b etending the positive ras to include negative values. Displaing all three aes with both positive and negative values requires tilting and rotating the aes slightl. 0 In the Cartesian coordinate sstem, the aes divide the space into quadrants. In the three-dimensional coordinate sstem, the aes divide space into eight octants. 0 Lesson. l Point Plotting in Three Dimensions 93

8 9. Draw several three-dimensional coordinate sstems. Draw each using a different orientation and label each ais. 9 Chapter l Three-Dimensional Coordinates

9 Problem Points in 3-D Space. Describe how to graph the point 3 on a number line.. Describe how to graph the point (, ) on the coordinate plane. 3. Describe how to graph the point (, 3, ) on the three-dimensional coordinate sstem.. Graph the point (, 3, ) Lesson. l Point Plotting in Three Dimensions 9

10 . Graph the points (,, ), (, 0, ), (0, 3, ), and (,, 0) Consider a coordinate plane. a. Describe the coordinates of all the points on the -ais. b. What is an equation for the -ais? c. Describe the coordinates of all the points on the -ais. d. What is an equation for the -ais? 7. Consider a three-dimensional coordinate sstem. a. Describe the coordinates of all points on the -plane. b. What is an equation of the -plane? c. Describe the coordinates of all the points on the -plane. d. What is the equation of the -plane? 96 Chapter l Three-Dimensional Coordinates

11 e. Describe the coordinates of all the points on the -plane. f. What is the equation of the -plane? Once an origin and -, -, and -aes are defined, an point can be represented as an ordered triple. Conversel, the point represented b an ordered triple can be uniquel located. So, the three-dimensional coordinate sstem full represents three-dimensional space. Be prepared to share our methods and solutions. Lesson. l Point Plotting in Three Dimensions 97

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13 . Planes Equations of Planes Objective In this lesson ou will: l Graph planes in three dimensions. Problem Equations with One Variable. Graph all points defined b on the number line.. Graph all points defined b on the coordinate plane. Lesson. l Equations of Planes 99

14 3. Graph all points defined b on the three-dimensional coordinate sstem.. Describe the graph of a. On a number line b. On a coordinate plane c. On the three-dimensional coordinate sstem 00 Chapter l Three-Dimensional Coordinates

15 . Graph each equation. a. b. 3 Lesson. l Equations of Planes 0

16 c Describe the graph of an equation that involves one variable in three dimensions. 0 Chapter l Three-Dimensional Coordinates

17 Problem Equations with Two Variables. Graph all points defined b on the coordinate plane.. Graph all points defined b on the three-dimensional coordinate sstem. 3. Describe the graph of a. On a coordinate plane b. On the three-dimensional coordinate sstem Lesson. l Equations of Planes 03

18 . Graph each equation. a. b. 0 Chapter l Three-Dimensional Coordinates

19 c.. Describe the graph of an equation that involves two variables in three dimensions. Problem 3 Equations with Three Variables. Consider the equation 3. a. Complete the table. 6 6 Lesson. l Equations of Planes 0

20 b. Calculate the -, -, and -intercepts of the graph. c. Graph the points from the table and the intercepts. d. Describe the graph of 3. e. Graph the equation Chapter l Three-Dimensional Coordinates

21 The definition of a function can be etended to three dimensions b defining a function with two independent variables, and, as f(, ). The equation from Question can be rewritten as the function f(, ) 3.. Graph each function using a table. a. f(, ) Lesson. l Equations of Planes 07

22 b. f(, ) c. f(, ) Chapter l Three-Dimensional Coordinates

23 d. f(, ) 3 6 Be prepared to share our methods and solutions. Lesson. l Equations of Planes 09

24 0 Chapter l Three-Dimensional Coordinates

25 .3 Distance Distance in Three Dimensions Objectives In this lesson ou will: l Derive the distance formula in three dimensions. l Calculate the distance between two points in space. l Derive the midpoint formula in three dimensions. l Determine a midpoint between two points in space. Ke Term l diagonal of a solid figure Problem Deriving the Distance Formula in Three Dimensions. Calculate the distance between each pair of points on a number line. a. and D Take Note The distance between two points on a number line is the absolute value of the difference of the two numbers. b. 6 and 0 D c. and 7 D d. 6 and 9078 D A second method to calculate the distance between two points on a number line is: l Calculate the difference between the points. l Square the difference. l Determine the principal square root. Lesson.3 l Distance in Three Dimensions

26 This method can be represented b the formula D (a b).. Calculate the distance between each pair of points in Question using this formula. a. and D b. 6 and 0 D c. and 7 D d. 6 and 9078 D 3. Do both methods for calculating the distance between two points on a number line ield the same results? Eplain wh or wh not.. Calculate the distance between two points (a, b) and (c, d ) in two dimensions b completing the following steps. (a, b) (c, d ) a. Plot the points and then draw a rectangle such that the two points are endpoints of a diagonal of the rectangle. b. Label the coordinates of the other two vertices of the rectangle. Chapter l Three-Dimensional Coordinates

27 c. Calculate the length of each side of the rectangle. d. Calculate the length of the diagonal using the Pthagorean Theorem.. How does the formula from Question (e) compare to the formula for calculating distance on a number line? 6. Calculate the distance between two points (a, b, c) and (d, e, f) in the three-dimensional coordinate sstem b completing the following steps. a. Plot the points. (a, b, c) (d, e, f ) b. Draw a rectangular prism such that the two points are endpoints of a diagonal of the prism. c. Label the coordinates of the other si vertices of the prism. d. Draw a diagonal of a face of the prism from (a, b, c) to a verte that shares an edge with (d, e, f ). e. Use the Pthagorean Theorem to calculate the length of this diagonal. Lesson.3 l Distance in Three Dimensions 3

28 f. Calculate the length of the diagonal of the prism using the length of an edge and the length of the diagonal from Question 6(e). 7. How does the formula from Question 6(f) compare to the formula for calculating distance on a number line, and calculating distance in the coordinate plane? Problem Calculating Distance in Three Dimensions. Calculate the distance between each pair of points. a. (, 3, ), (, 0, 0) b. (,, 7), (, 7, 9) c. (0, 0, 0), (0, 3, 3) d. (, 3, ), (,, 0) Chapter l Three-Dimensional Coordinates

29 . Calculate each unknown coordinate. a. The distance between the points (, 3, ) and (,, 0) is 0. b. The distance between the points (,, ) and (,, 3) is. Problem 3 Deriving the Midpoint Formula in Three Dimensions To build models, such as the Sierpinski Tetrahedrons shown, ou must be able to determine the midpoint of each side of the tetrahedron. Is there a formula for calculating the midpoint in three dimensions? Lesson.3 l Distance in Three Dimensions

30 . Consider points A( ) and B() on a number line. A B a. Locate the point on the number line that is half wa between points A and B. What is the value of this point? b. Two points on a number line have the values of and. How can ou calculate the value of the point half wa between these two points?. Consider points A(, ) and B( 3, 3) on the coordinate plane. Connect points A and B to form line segment AB. 8 6 B A a. Calculate the coordinates of the midpoint of line segment AB. b. Consider points A(, ) and B(, ) on the coordinate plane. Calculate the coordinates of the midpoint of line segment AB. 6 Chapter l Three-Dimensional Coordinates

31 The formula for determining the midpoint between the values and on a number line (one dimension) is. The formula for determining the coordinates of the midpoint of a line segment on the coordinate plane (two dimensions) is (, ). You will use a familiar solid to derive the coordinates of a midpoint in three dimensions. A diagonal of a solid figure is a line segment with one endpoint at a verte and another endpoint at a verte that does not share an edge or a face with the other endpoint. 3. Derive the coordinates of the midpoint of the diagonals of the rectangular prism shown. All coordinates are real numbers. (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) (,, ) a. Connect point (,, ) to point (,, ). What relationship does this line segment have to the base of the rectangular prism? Lesson.3 l Distance in Three Dimensions 7

32 b. Connect point (,, ) to point (,, ). What relationship does this line segment have to the rectangular prism? c. A right triangle is formed b the diagonals drawn in part (a), part (b), and an edge of the rectangular prism. What are the coordinates of the vertices of this right triangle? d. Eplain how ou would locate the midpoint of the diagonal drawn in part (a). e. What are the coordinates of the midpoint of the diagonal drawn in part (a)? 8 Chapter l Three-Dimensional Coordinates

33 f. Eplain how ou would locate the midpoint of the edge of the rectangular prism used to form the right triangle. g. What are the coordinates of the midpoint of the edge of the rectangular prism used to form the right triangle? h. Eplain how ou would locate the midpoint of the diagonal drawn in part (b). i. What are the coordinates of the midpoint of the diagonal drawn in part (b)? Lesson.3 l Distance in Three Dimensions 9

34 j. Imagine three planes intersecting at the midpoint of the diagonal drawn in part (b). Describe the coordinate that is the same across each plane. -plane: -plane: -plane: k. Write a formula to determine the midpoint of a line segment with endpoints (,, ) and (,, ) where,,,,, and are real numbers.. Consider points A(,, ) and B( 3, 0, ) on the three-dimensional coordinate sstem. Calculate the coordinates of the midpoint of line segment AB. B A 3 Be prepared to share our methods and solutions. 0 Chapter l Three-Dimensional Coordinates

35 . Volume in Space Models and Volume Objectives In this lesson ou will: l Model three-dimensional solids on a coordinate sstem. l Determine the vertices of three- dimensional figures. l Calculate volumes of solid figures. l Calculate the coordinates of a midpoint in three dimensions. Ke Term l regular tetrahedron Problem Coordinate Geometr in Three Dimensions The two-dimensional coordinate plane is useful to eplore the properties of geometric figures such as triangles and quadrilaterals. Similarl, the three-dimensional coordinate sstem is useful to eplore the properties of solid figures. Consider a rectangular prism whose dimensions are 3 units units units. The base of the rectangular prism is drawn on the -plane.. Complete the drawing of the rectangular prism. Label each verte with a capital letter Lesson. l Models and Volume

36 . Identif the coordinates of each verte. 3. Name the diagonals of the rectangular prism.. Calculate the length of each diagonal.. What do ou notice about the lengths of each diagonal? 6. Describe other was to draw the rectangular prism in the same octant. Chapter l Three-Dimensional Coordinates

37 7. Sketch each of the views that ou described in Question Lesson. l Models and Volume 3

38 8. Prove that the lengths of the diagonals of an right rectangular prism are equal b completing the following steps. a. Draw a right rectangular prism in the first octant with dimensions a, b, and c. b. Label each verte with a capital letter. Identif the coordinates of each verte. c. Calculate the length of each diagonal. Chapter l Three-Dimensional Coordinates

39 Problem Pramids and Tetrahedrons. A right square pramid has an altitude of 6 units. The length of each edge of the base is units. a. Draw the base of this pramid on the -plane b. Label each verte with a capital letter. Identif the coordinates of each verte of the base. c. Identif two possible coordinates of the remaining verte. Lesson. l Models and Volume

40 d. Draw the pramid formed b each verte from Question (c) e. Calculate the midpoint of one edge of the base of the pramid. f. Calculate the slant height of the pramid. 6 Chapter l Three-Dimensional Coordinates

41 A regular tetrahedron is a solid that has four faces that are equilateral triangles.. The length of each edge of a regular tetrahedron is units. a. Draw the regular tetrahedron in the first octant with one edge on the -ais b. Label each verte with a capital letter. Identif the coordinates of each verte on the -plane. Lesson. l Models and Volume 7

42 c. Calculate the coordinates of the remaining verte. d. Calculate the height of the tetrahedron. e. Calculate the volume of the tetrahedron. 8 Chapter l Three-Dimensional Coordinates

43 3. Derive a formula for the volume of a tetrahedron with side length. Lesson. l Models and Volume 9

44 Problem 3 Midpoints in Three Dimensions. Calculate the coordinates of the midpoint of each edge of the rectangular prism in Problem. 30 Chapter l Three-Dimensional Coordinates

45 . Calculate the coordinates of the midpoint of each edge of the right square pramid in Problem. a. Use the vertices of the base from Problem, Question (b) and the first possible coordinate for the remaining verte from Problem, Question (c). Lesson. l Models and Volume 3

46 b. Use the vertices of the base from Problem, Question (b) and the second possible coordinate for the remaining verte from Problem, Question (c). Be prepared to share our methods and solutions. 3 Chapter l Three-Dimensional Coordinates