UNIT 6 Three-Dimensional Figures and Graphs -...

Unit 6 three-dimensional figUres and graphs 173

UNIT 6 Three-Dimensional Figures and Graphs

We live in a three-dimensional world that is often represented on a two-dimensional plane. drawings, paintings, and maps are some of the two-dimensional representations of our world.

artists were not always able to accurately represent three dimen-sions on a flat surface. in the fifteenth century, filippo Brunelleschi, an italian sculptor and architect with an interest in mathematics, was the first to conduct experiments that led to an understanding of one-point perspective. to create the illusion of depth on a two-dimensional surface, he demonstrated that lines that are parallel in the real world should be drawn as meeting at a single vanishing point. in this unit, you will extend your geometric knowledge and skills to three-dimensional figures.

UNIT OBJECTIVES

► draw basic three-dimensional figures as well as two-dimensional views of three-dimensional figures.

► define polyhedron and identify the faces, edges, and vertices of polyhedra.

► define prism and the parts of a prism.

► Classify prisms.

► find the surface area of prisms.

► find the volume of prisms.

► locate and plot points in a three-dimensional coordinate system.

► Use the distance and midpoint formulas for three dimensions.

► Use intercepts to graph planes in space.

► Use parametric equations to plot lines in space.

Copyright © 2006, K12 Inc. All rights reserved. This material may not be reproduced in whole or in part, including illustrations, without the express prior written consent of K12 Inc.

solid shapes and three-dimensional draWing 175

Solid Shapes and Three-Dimensional Drawing

OBJECTIVES During the Renaissance, artists started using perspective to represent the three-dimensional world on their two-dimensional canvases. Notice that the painting on the right has depth and looks more realistic than the one on the left.

► Define three-dimensional drawing and depth.

► Use isometric grid paper to show how three-dimensional figures can be drawn.

► Draw two-dimensional views of three-dimensional figures.

► Interpret a three-dimensional drawing.

► Solve problems involving surface area and volume.

KEYWORDS depth isometric drawingorthographic views perspective drawingsolids surface area three-dimensional drawing volume

Figures in Three Dimensions

Polygons are two-dimensional figures. They are flat and lie on a single plane. Solids are three-dimensional figures. Three-dimensional figures have the additional dimension of depth. Depth is a measure of the length of a figure from the front of the figure to the back.


176 Unit 6 three-dimensional figUres and graphs

vanishing point horizon

1-point perspective 2-point perspective

Drawing Three-Dimensional FiguresA three-dimensional drawing represents a three-dimensional figure on a two-dimensional plane. The three-dimensional drawings shown below are called perspective drawings because the objects appear true-to-life, as they would if you were actually looking at them.

You can draw a cube in one-point perspective by first drawing a square, a line called the horizon, and a point on the horizon called the vanishing point. This square is the front of the cube. Draw straight lines from the vertices of the square to the vanishing point. Draw the back face of the cube so that its vertices are on the lines leading to the vanishing point. You may find it easiest to draw the top horizontal segment first. Because it is a cube, locate the points so that all sides appear to be the same length. Connect the front and back squares by tracing over the segments between them.

To draw a cube in two-point perspective, start with the horizon, two vanishing points, and a vertical line segment that will be the front edge of the cube. Draw lines from the front segment’s endpoints to each vanishing point and use those to draw the left and right vertical sides of the cube. Join the top of each edge to its opposite vanishing point to form the top face of the cube.

Look carefully at the sides and angles of each cube in the diagrams. Some sides that should be parallel are not drawn parallel, and some angles do not actually measure 90°.

Isometric Drawings of Three-Dimensional FiguresAnother type of three-dimensional drawing is called an isometric drawing. Isometric drawings show three sides of a solid figure from a corner view. Notice in the diagram at left that the parallel lines are drawn parallel and the right angles on the top face are drawn as 120° and 60° angles. This is one way isometric drawings differ from perspective drawings.

You can use isometric grid paper to help you draw three-dimensional figures.


solid shapes and three-dimensional draWing 177

1-point perspective

top bottom left side right side front back

set of orthographic views

top bottom left side right side front back

Orthographic Views Orthographic views of a three-dimensional figure show the front, back, top, bottom, left side, and right side views of the figure. The orthographic views of a square pyramid are shown in the diagram.

To draw a top view, imagine flying above the object and looking straight down on it. If you flew above the pyramid and looked down, you would see more than just the top point. You would see the outline of the base of the fig-ure as well as the four edges that lead to the corners of the base. For a bottom view, imagine lying below the object and looking straight up. For side views, imagine floating around the figure and recording what you see when you face each side directly.

The orthographic views of a previous drawing are shown below. Notice that in the left side, right side, front, and back views, the depth of each block is irrelevant. If you can see a block from that side, regardless of its depth, a square is included in the orthographic view.

Just a couple of views of a figure do not tell you what the figure looks like. Though three views are sufficient for some figures, other figures require as many as six views to get an accurate view of what the figure looks like.

isometric drawing

set of orthographic views



RECONNECT TO THE BIG IDEA

5 in.

3 in.

12 in.

4 cm

3 cm 5 cm

Surface Area and Volume The surface area of a figure is the amount of space that covers the figure. To find the surface area of a solid, find the sum of the areas of its outer surfaces.

Compare the area calculations below with the three-dimensional figure shown at left.

top 12 × 5 = 60bottom 12 × 5 = 60left side 3 × 5 = 15right side 3 × 5 = 15 front 12 × 3 = 36

+ back 12 × 3 = 36 222

The surface area of the three-dimensional figure is 222 in2.

The volume of a solid is the amount of space inside the figure, measured in cubic units. If it’s a small rectangular figure, you may be able to count the cubes. This is normally not possible, however, because of either the size or the shape of the figure.

In a three-dimensional figure like the prism at left, you can multiply or count to find the area of one “layer” of cubes and multiply by the number of layers. Because each layer in this figure is a rectangle, you can find its volume by multiplying its dimensions.

V = 3 × 5 × 4 = 60

The volume of the figure is 60 cm3.

Summary

• Perspectivedrawingsshowthree-dimensionalobjectsas theyappear inreal life. To draw in perspective, use a horizon and vanishing points.

• Isometric drawings show three sides of a solid figure from a cornerview.

• Orthographicviewsofathree-dimensionalfigureshowthefront,back,top, bottom, left side, and right side views of the figure.

• Thesurfaceareaofasolidistheamountofspacethatcoversthefigure.• Thevolumeofasolidistheamountofspaceinsidethefigure.

Remember Measurement is the process of using a unit to determine how much of something you have.


lines, planes, and polyhedra 179

Lines, Planes, and Polyhedra

half-plane

OBJECTIVES A globe is a fair representation of the earth—it is a three-dimensional model representing a three-dimensional shape. However, flat maps of the earth, called map projections, distort the earth’s shape. Projecting a curved figure onto a flat surface always creates some type of deformation. Unlike the shape of the earth, many other three-dimensional shapes can be created accurately from two-dimensional models.

► Identify and define skew lines, half-planes, and dihedral angles.

► Define polyhedron.

► Identify the faces, edges, and vertices of polyhedra.

► Create a figure from its net, and draw nets of polyhedra.

KEYWORDS dihedral angle edge face half-planenet parallel planespolyhedron skew linesvertex of a polyhedron

Lines and Planes in Space

If two lines lie on the same plane, then those lines either intersect or are parallel. If two lines lie in different planes, there is another possibility. The lines can be skew. Skew lines are two lines that do not intersect but are not parallel. The blue lines in the diagram are skew lines.

Parallel planes are planes that do not intersect. For instance, the top and bottom faces of a cube lie on parallel planes.

A line that lies in a plane separates the plane into two half-planes. A half-plane consists of all the points on either side of the line.



RemembeR

dihedral angles

rectangular prism6

128

facesedgesvertices

square pyramid585

facesedgesvertices

octahedron8

126

facesedgesvertices

net of a square pyramid

If two planes intersect, they form four angles and four half-planes. Each angle is called a dihedral angle. A dihedral angle is formed by two non-coplanar half-planes and their line of intersection.An angle is two rays joined at

a point. A dihedral angle is two half-planes joined at a line.

PolyhedraA solid is a three-dimensional figure. A polyhedron is a solid enclosed by polygons. So, although a sphere is a solid, it is not a polyhedron. Note that the two plurals of polyhedron are polyhedra and polyhedrons.

The flat surfaces of a polyhedron are called the faces. The faces meet to form the edges of the polyhedron. The vertices of the faces meet to form the vertices of the polyhedron. The number of faces, edges, and vertices of three polyhedra are shown below.

Nets of PolyhedraA net is what a solid looks like if you unfold it. You can think of a net as a pattern for making a solid, much in the same way that a pattern is used to sew pieces of clothing together. Nets are often used to find surface area.


lines, planes, and polyhedra 181

To create the net of an octahedron with geometry software, create an equi-lateral triangle, and then use rotations and reflections to create the net in the diagram.Ifyoulike,youcanaddcolor.Onceyouhavecreatedyournet,youcan print it and fold it to create the polyhedron.

You can use grid paper to create a net. To make sure the figure folds up properly, be sure the top and bottom faces are congruent and that the height for the front and back is the same as for the top and bottom.

Summary

• Skewlinesdonotlieinthesameplaneanddonotintersect.• Parallelplanesareplanesthatdonotintersect.• Alineinaplaneseparatestheplaneintotwohalf-planes.• Anangle formedby the intersectionof twoplanes is called adihedral

angle.• Apolyhedronisasolidenclosedbypolygons.Eachpolygoniscalleda

face. The faces intersect at edges. The vertices of the faces meet to form the vertices of the polyhedron.

• Anetshowswhatasolidlookslikewhenthesolidhasbeenunfolded.Anet can also serve as a pattern for a solid.


prisms 183

Prisms

lateral face

bases

hexagonal prism

OBJECTIVES In the picture, a triangular prism is dispersing light into all of its different col-ors. Prisms that reflect, refract, and disperse light are called optical prisms. Opticalprismscomeinmanyshapesandsizes.Youwilllearnaboutvariouskinds of prisms in geometry and how to measure them.

► Define prism and the parts of a prism.

► Classify prisms.

► Solve problems that involve the diagonal of a right prism.

► Find the surface area and volume of prisms.

KEYWORDS base area of a prism diagonal of a polyhedronlateral edge lateral faceoblique prism prismright prism

Types of Prisms

A prism is a polyhedron with two parallel, congruent faces called bases. We call the other faces lateral faces. Notice that lateral faces are parallelograms formed by parallel segments that connect corresponding vertices of the bases. These parallel segments are called the lateral edges. Prisms are classified by the shapes of their bases. If the bases of a prism are triangles, we call it a triangular prism; if the bases are hexagons, we call it a hexagonal prism, and so on.



oblique triangular prism

right rectangular prism

8 cm

10 cm4 cm

If a prism is not a right prism, it is an oblique prism. The lateral faces of an oblique prism are not perpendicular to the bases.

Diagonals A diagonal of a polyhedron is a line segment that joins two vertices that are in different faces. Like diagonals of polygons, they do not overlap the sides of the figure itself. A diagonal of a right rectangular prism is shown in red at left.

Formula for the Diagonal of a Right Rectangular Prism

The length of a diagonal d of a right rectangular prism is

d = √__________

l 2 + w2 + h2

For the right rectangular prism at left:

d = √____________

102 + 82 + 42

= √_____________

100 + 64 + 16

= √____

180

≈ 13.42

The length of the diagonal is approximately 13.42 cm.

Surface Area of PrismsRecall that surface area tells how much space covers a figure. The surface area of a prism is the sum of the areas of the bases and the lateral faces. The area of one base is defined as B. The sum of the areas of the lateral faces is defined as L. Because the bases are congruent, we write the sum of the bases as 2B.

A prism whose lateral edges are perpendicular to the bases is a right prism.


prisms 185

RemembeR

4 in.

3 in.10 in.

5 in.

Formula for the Surface Area of a Prism

The surface area S of a prism, where B is the area of one base and L is the sum of the areas of the lateral faces, is

S = 2B + L

Follow the steps to find the surface area of the right triangular prism below.

The lateral edges are all congruent.

Step 1: Find B and 2B by using the formula for the area of a triangle.

A = 1 __ 2 bh

= 1 __ 2 (3)(4)

= 6

The area of one base is 6 in2. So 2B is 2(6 in2) = 12 in2.

Step 2: Find L. There are three lateral faces.

A = bh A = bh A = bh

= 10(3) = 10(4) = 10(5)

= 30 = 40 = 50

L = 30 + 40 + 50 = 120

Step 3: Find S.

S = 2B + L

= 12 + 120

= 132

The surface area of the prism is 132 in2.

Volume of PrismsRecall that volume is the amount of space inside a figure. The volume of a prism can be found by multiplying the area of a base B by its height h. The height of a prism is the perpendicular distance between the bases. When the prism is oblique, the height will be different from the length of a lateral edge.



4 in.

3 in.10 in.

5 in.

Formula for the Volume of a Prism

The volume V of a prism, where B is the area of one base and h is the height of the prism, is

V = Bh

To find the volume of the same right triangular prism for which you just found the surface area, see the following steps.

Step 1: Find B.

A = 1 __ 2 bh

= 1 __ 2 (3)(4)

= 6

The area of a base is 6 in2.

Step 2: Identify h. The height is the distance between the bases. In this figure it is 10 inches.

Step 3: Find V.

V = Bh = 6(10) = 60

The volume of the prism is 60 in3.

Summary

• Aprism isapolyhedronwith twoparallel,congruentbasesand lateralfaces that are parallelograms formed by the parallel segments that connect corresponding vertices of the bases.

• A prismwhose lateral edges are perpendicular to the bases is a rightprism. A prism whose lateral edges are not perpendicular to the bases is an oblique prism.

• Thediagonalofapolyhedronisalinesegmentthatjoinstwoverticesthatare in different faces.

• Thelengthofadiagonald of a right rectangular prism can be found by using the formula d = √

__________ l 2 + w2 + h2 .

• ThesurfaceareaS of a prism, where B is the area of one base and L is the sum of the areas of the lateral faces, is S = 2B + L.

• ThevolumeV of a prism, where B is the area of one base and h is the height of the prism, is V = Bh.

RemembeRArea is measured in square units. Volume is measured in cubic units.


Coordinates in three dimensions 187

Coordinates in Three Dimensions

y

x-10 -8 -4 -2 2

(3, 6)

4 6 8 10-2

2468

10

-4-6-8

-10

-6

y

x

OBJECTIVES Architects, interior designers, and radiologists are just a few of the professionals that have benefited from three-dimensional computer imaging technology. Architects use computer software to create three-dimensional computer models of buildings. Interior designers create virtual rooms where a client can change wall color or floor tiling with the click of a mouse. Radiologists use computer imaging to “see” inside the human body, using a three-dimensional view, and are able to detect medical conditions at a much earlier stage. This lesson introduces you to a three-dimensional coordinate system.

KEYWORDS coordinate plane first octant octant ordered tripleright-handed system three-dimensional coordinate system

The Three-Dimensional Coordinate System

The coordinate plane you are familiar with is formed by the intersection of an x-axis and a y-axis. Each point in the coordinate plane is represented by an ordered pair of real numbers (x, y). You can locate and plot a point by first starting at the origin, then moving left or right according to the x-coordinate, and finally moving up or down according to the y-coordinate. Remember that the positive x-direction points to the right, and the positive y-direction points upward.

Plotting points in three dimensions can be accomplished using a simi-lar method. First, we must add a third axis to the system, the z-axis, which will correspond to the third dimension. To visualize the addition of this axis, imagine the ordinary xy-plane lying flat on a table, with the positive x-direc-tion pointing toward you and the positive y-direction pointing to the right, as shown in the figure.

► Identify characteristics of a three-dimensional coordinate system.

► Locate and plot points in a three-dimensional coordinate system.

► Use the Distance and Midpoint Formulas for three dimensions.



y

x

z

(0, 0, 0)

x

z

y

1

4 8

56

2

3

7

zy

x

Now imagine a third axis, the z-axis, which is perpendicular to both the x- and y-axes. The positive direction for this new axis will point upward. By adding this axis, we have formed a three-dimensional coordinate system.

Notice that in forming this coordinate system, we could have chosen the positive z-direction to point downward. By instead choosing it to point upward, we have defined a right-handed system. This name comes from the so-called “right-hand rule”: If you imagine wrapping your right hand around the z-axis, with your fingers curling from the positive x-direction axis toward the positive y-direction, as shown in the figure, then your thumb will point in the positive z-direction.

Had we chosen the positive z-direction to point downward, we would have formed a left-handed system. In most applications, however, right-handed systems are used.

In a three-dimensional coordinate system, the intersection of the x-, y-, and z-axes creates three intersecting planes, the xy-, yz- and xz-planes. These planes divide the system into eight parts. Each part is called an octant. In the first octant the values of x, y, and z are all positive.

Each point in a three-dimensional coordinate system is associated with a unique ordered triple of real numbers having the form (x, y, z). Given an ordered triple, you can locate and plot the corresponding point using the following method. First, start at the origin. Then move forward or backward according to the x-coordinate. Next, move left or right according to the y-coordinate. Finally, move up or down according to the z-coordinate.


Coordinates in three dimensions 189

(0, 0, 0)

3 units

4 units

3 units

(0, 0, 0)

z

y

x4 units

z

y

x

(0, 0, 0)

3 units

4 units

z

y

x

(3, 4, 4)

For example, to plot the point (3, 4, 4), begin at the origin, and then move 3 units in a positive direction (forward) along the x-axis. Then move 4 units in a positive direction (right) along the y-axis. To help show perspective, you can draw a rectangle in the xy-plane having one corner at the origin and another corner at (3, 4, 0). Finally, after drawing the rectangle, move 4 units in a positive direction (up) along the z-axis.

Finding the Distance Between PointsFinding the distance between two points in space is similar to finding the distance between two points on a plane.

Distance Formula for Three Dimensions

The distance d between the points (x1, y1, z1) and (x2, y2, z2) in space is

d = √_________________________

(x2 – x1)2 + (y2 – y1)

2 + (z2 – z1)2

The following is how you would find the distance between (4, 4, 3) and (5, 0, 6):

d = √_________________________

(x2 – x1)2 + (y2 – y1)

2 + (z2 – z1)2

= √________________________

(5 – 4)2 + (0 – 4) 2 + (6 – 3)2

= √_________________

(1)2 + (– 4) 2 + (3)2

= √__________

1 + 16 + 9

  = √___

26

≈ 5.10



Finding the Midpoint of a Line SegmentThe formula for the midpoint of a line segment in space is an extension of the formula for the midpoint of a line segment in the coordinate plane.

Midpoint Formula in Three Dimensions

The midpoint of a segment with endpoints (x1, y1, z1) and (x2, y2, z2) in space has these coordinates:

( x1 + x2 ______ 2 , y1 + y2 ______ 2 ,

z1 + z2 ______ 2 ) The following is how you would find the midpoint of a line segment with endpoints (8, 5, 2) and (4, 6, 0):

( x1 + x2 ______ 2 , y1 + y2 ______ 2 ,

z1 + z2 ______ 2 ) ( 8 + 4 _____ 2 , 5 + 6 _____ 2 , 2 + 0 _____ 2 )

( 6, 5 1 __ 2 , 1 )

Summary

• Acoordinatesystemhasthreeaxes—anx-axis, a y-axis, and a z-axis.• Pointsinspacearenamedbytheuseoforderedtriples.• Theaxesformanxy-plane, a yz-plane, and an xz-plane, which divide the

coordinate system into octants. • To find the distance between two points in space, (x1, y1, z1) and

(x2, y2, z2), use the Distance Formula for Three Dimensions:

d = √_________________________

(x2 – x1)2 + (y2 – y1)

2 + (z2 – z1)2 .

• Thecoordinatesofthemidpointofalinesegmentinspacewithendpoints

(x1, y1, z1) and (x2, y2, z2), are ( x1 + x2 ______ 2 , y1 + y2 ______ 2 ,

z1 + z2 ______ 2 ) .


eqUations of lines and planes in spaCe 191

Equations of Lines and Planes in Space

OBJECTIVES When Lydia went hiking, she took her Global Positioning System (GPS) device and marked the parking lot as her first waypoint. After a few hours of hiking up steep hills and across streams, she realized that she didn’t know the way back. She looked at her GPS device, chose the first waypoint, and selected “Go to.” Easy! The only problem came when the route took her off the trail and straight into some poison ivy. The GPS device determined a line between Lydia, who was standing on a steep hill, and the parking lot. If she could imagine the route from her point on the hill down to the parking lot, she would be imagining a line in space.

► Use intercepts to graph a plane in space.

► Define trace and find the equations of traces given the equation of a plane.

► Use parametric equations to plot lines in space.

► Define the equation of a plane in space.

KEYWORDS interceptparametric equationtrace

Equations in Space

Onthecoordinateplane,aninterceptiswherealinecrossesanaxis.Inspace,an intercept is a point where a plane crosses an axis. You can use the x-, y-, and z-intercepts to help you sketch a plane in space.



z

x

y

(0, 0, 3)

O

(6, 0, 0)

(0, 2, 0)

(10, 0, 0)

(0, 0, 2)(0, 5, 0)

z

y

x

Equation of a Plane in Space

Given that A, B, C, and D are real numbers not all equal to zero and A is nonnegative, the equation of a plane in space is

Ax + By + Cz = D

Let us sketch the plane 2x + 6y + 4z = 12.

To find the x-intercept, substitute 0 for y and z.

2x + 6(0) + 4(0) = 12 2x = 12 x = 6

• Thex-intercept is the point (6, 0, 0).

To find the y-intercept, substitute 0 for x and z.

2(0) + 6y + 4(0) = 12 6y = 12 y = 2

• They-intercept is the point (0, 2, 0).

To find the z-intercept, substitute 0 for x and y.

2(0) + 6(0) + 4z = 12 4z = 12 z = 3

• Thez-intercept is the point (0, 0, 3).

To sketch the plane, plot the intercepts, draw the triangle connecting these points, and shade the plane. Remember—a plane extends forever on all sides.

Similarly, if you have the graph of a plane, you can work backward to find an equation from its intercepts. For example, consider the plane with an x-intercept of (10, 0, 0), a y-intercept of (0, 5, 0), and a z-intercept of (0, 0, 2). Each of these points gives values of x, y, and z that satisfy the equation Ax + By + Cz = D. Substitute these values into the equation and solve for A, B, C, and D.

Substituting 10 for x, 0 for y, and 0 for z gives: A(10) + B(0) + C(0) = D 10A = D

A = D ___ 10

Substituting 0 for x, 5 for y, and 0 for z gives: A(0) + B(5) + C(0) = D 5B = D

B = D __ 5



(4, 0, 0)

(0, 0, 1)

z

y

x

z

y

x

(0, 0, 5)

Substituting 0 for x, 0 for y, and 2 for z gives: A(0) + B(0) + C(2) = D 2C = D

C = D __ 2

When we put these values back into Ax + By + Cz = D we get:

D ___ 10 x + D __ 5 y + D __ 2 z = D

Because the plane does not pass through the origin, D is not zero, so we can divide both sides by D.

x ___ 10 + y __ 5 + z __ 2 = 1.

Onceyoumultiplytheequationbytheleastcommondenominatorof10,youhave the equation in standard form, where A, B, C, and D are all integers.

x + 2y + 5z = 10

The plane graphed below has an x-intercept at (4, 0, 0) and a z-intercept at (0, 0, 1), but it has no y-intercept. This plane is parallel to the y-axis.

You can still find an equation for this plane by eliminating the y term from the equation Ax + By + Cz = D and solving as before. The equation becomes Ax + Cz = D. Substituting (4, 0, 0) and (0, 0, 1) gives:

A(4) + C(0) = D A(0) + C(1) = D 4A = D C = D

A = D __ 4

So x __ 4 + z = 1, or x + 4z = 4, is an equation for this plane.

Finally, consider the plane shown below. It is perpendicular to the z-axis, so it has neither an x-intercept nor a y-intercept, but it has a z-intercept at (0, 0, 5).



z

yz-trace

xy-trace

xz-tracey

x

In this case, eliminate both the x- and y-terms from Ax + By + Cz = D. This gives Cz = D. Substituting (0, 0, 5) gives 5C = D, which gives z __ 5 = 1. This is the same as z = 5.

Equations of TracesYou learned previously that the intersection of the three axes creates three coordinate planes, the xy-, yz-, and xz-planes. A trace is the intersection of a plane with one of the coordinate planes. Remember that two planes intersect in a line.

• An xy-trace is the line formed by the intersection of a plane with the xy-plane. The points on an xy-trace all have a z-coordinate of 0, so to find the equation of the trace, you set z to 0 in the equation of the plane.

• A yz-trace is the line formed by the intersection of a plane with the yz-plane. The points on a yz-trace all have an x-coordinate of 0, so to find the equation of the trace, you set x to 0 in the equation of the plane.

• An xz-trace is the line formed by the intersection of a plane with the xz-plane. The points on an xz-trace all have a y-coordinate of 0, so to find the equation of the trace, you set y to 0 in the equation of the plane.

Suppose you want to find the equations of the traces for the plane given by 5x + 6y – 2z = 30.

To find the xy-trace, substitute 0 for z.

5x + 6y – 2(0) = 30 5x + 6y = 30

• Theequationofthexy-trace is 5x + 6y = 30.

To find the yz-trace, substitute 0 for x.

5(0) + 6y – 2z = 30 6y – 2z = 30 3y – z = 15

• Theequationoftheyz-trace is 3y – z = 15.

To find the xz-trace, substitute 0 for y.

5x + 6(0) – 2z = 30 5x – 2z = 30

• Theequationofthexz-trace is 5x – 2z = 30.



Parametric Equations and Equations of LinesParametric equations express variables in terms of another variable, called the parameter. The variable t is usually the variable used to represent the parameter. In this book, parametric equations will be used to describe the equation of a line in three dimensions. Later in your math and science studies, you may use parametric equations to represent time and motion.

Suppose we want to graph the line whose parametric equations are as follows:

x = 6t + 1 y = t – 1 z = 3t + 1

First, make a table and choose values for t.

t x y z ordered triple

012

Second, substitute a value into each parametric equation to find the corresponding values for x, y, and z.

When t = 0,

x = 6(0) + 1 y = (0) – 1 z = 3(0) + 1 = 1 = –1 = 1

When t = 1,

x = 6(1) + 1 y = (1) – 1 z = 3(1) + 1 = 7 = 0 = 4

When t = 2,

x = 6(2) + 1 y = (2) – 1 z = 3(2) + 1 = 13 = 1 = 7

t x y z ordered triple

0 1 –1 1 (1, –1, 1)1 7 0 4 (7, 0, 4)2 13 1 7 (13, 1, 7)

Finally, plot the points (1, –1, 1), (7, 0, 4), and (13, 1, 7) and draw the line connecting them.



(13, 1, 7)

(7, 0, 4)

(1, –1, 1)

z

y

x

Summary

• Inspace,aninterceptisapointwhereaplanecrossesanaxis.Thex-, y-, and z-intercepts can help you sketch a plane.

• TheequationofaplaneisAx + By + Cz = D, where A, B, C, and D are real numbers not all equal to zero and A is nonnegative.

• Atraceistheintersectionofaplanewithoneofthecoordinateplanes.• Parametricequationsexpressvariablesintermsofanothervariablecalled

the parameter.• Youcanuseparametricequationstographalineinspace.


UNIT 6 Three-Dimensional Figures and Graphs -...

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