VECTORS AND THE GEOMETRY OF SPACE
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VECTORS AND THE GEOMETRY OF SPACE







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12. VECTORS AND THE GEOMETRY OF SPACE. VECTORS AND THE GEOMETRY OF SPACE. In this chapter, we introduce vectors and coordinate systems for three-dimensional space. VECTORS AND THE GEOMETRY OF SPACE.
VECTORS AND THE GEOMETRY OF SPACE

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Slide 1

12

VECTORS AND

THE GEOMETRY OF SPACE

Slide 2

VECTORS AND THE GEOMETRY OF SPACE

  • In this chapter, we introduce vectors and coordinate systems for three-dimensional space.

Slide 3

VECTORS AND THE GEOMETRY OF SPACE

  • This will be the setting for our study of the calculus of functions of two variables in Chapter 14.

    • This is because the graph of such a function is a surface in space.

Slide 4

VECTORS AND THE GEOMETRY OF SPACE

  • We will see that vectors provide particularly simple descriptions of lines and planes in space.

Slide 5

VECTORS AND THE GEOMETRY OF SPACE

12.1

Three-Dimensional Coordinate Systems

  • In this section, we will learn about:

  • Aspects of three-dimensional coordinate systems.

Slide 6

TWO-DIMENSIONAL (2-D) COORDINATE SYSTEMS

  • To locate a point in a plane, two numbers are necessary.

    • We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers—where a is the x-coordinate and b is the y-coordinate.

    • For this reason, a plane is called two-dimensional.

Slide 7

THREE-DIMENSIONAL (3-D) COORDINATE SYSTEMS

  • To locate a point in space, three numbers are required.

    • We represent any point in space by an ordered triple (a, b, c) of real numbers.

Slide 8

3-D COORDINATE SYSTEMS

  • In order to represent points in space, we first choose:

    • A fixed point O (the origin)

    • Three directed lines through O that are perpendicular to each other

Slide 9

COORDINATE AXES

  • The three lines are called the coordinate axes.

  • They are labeled:

    • x-axis

    • y-axis

    • z-axis

Slide 10

COORDINATE AXES

  • Usually, we think of:

    • The x- and y-axes as being horizontal

    • The z-axis as being vertical

Slide 11

COORDINATE AXES

  • We draw the orientation of the axes as shown.

Slide 12

COORDINATE AXES

  • The direction of the z-axis is determined by the right-hand rule, illustrated as follows.

Slide 13

COORDINATE AXES

  • Curl the fingers of your right hand around the z-axis in the direction of a 90° counterclockwise rotation from the positive x-axis to the positive y-axis.

    • Then, your thumb points in the positive direction of the z-axis.

Slide 14

COORDINATE PLANES

  • The three coordinate axes determine the three coordinate planes.

    • The xy-plane contains the x- and y-axes.

    • The yz-plane contains the y- and z-axes.

    • The xz-plane contains the x- and z-axes.

Slide 15

OCTANTS

  • These three coordinate planes divide space into eight parts, called octants.

    • The firstoctant, in the foreground, is determined by the positive axes.

Slide 16

3-D COORDINATE SYSTEMS

  • Many people have some difficulty visualizing diagrams of 3-D figures.

  • Thus, you may find it helpful to do the following.

Slide 17

3-D COORDINATE SYSTEMS

  • Look at any bottom corner of a room and call the corner the origin.

Slide 18

3-D COORDINATE SYSTEMS

  • The wall on your left is in the xz-plane.

  • The wall on your right is in the yz-plane.

  • The floor is in the xy-plane.

Slide 19

3-D COORDINATE SYSTEMS

  • The x-axis runs along the intersection of the floor and the left wall.

  • The y-axis runs along that of the floor and the right wall.

Slide 20

3-D COORDINATE SYSTEMS

  • The z-axis runs up from the floor toward the ceiling along the intersection of the two walls.

Slide 21

3-D COORDINATE SYSTEMS

  • You are situated in the first octant.

  • You can now imagine seven other rooms situated in the other seven octants.

    • There are three on the same floor and four on the floor below.

    • They are all connected by the common corner point O.

Slide 22

3-D COORDINATE SYSTEMS

  • Now, if P is any point in space, let:

    • a be the (directed) distance from the yz-plane to P.

    • b be the distance from the xz-plane to P.

    • c be the distance from the xy-plane to P.

Slide 23

3-D COORDINATE SYSTEMS

  • We represent the point P by the ordered triple of real numbers (a, b, c).

  • We call a, b, and c the coordinatesof P.

    • a is the x-coordinate.

    • b is the y-coordinate.

    • c is the z-coordinate.

Slide 24

3-D COORDINATE SYSTEMS

  • Thus, to locate the point (a, b, c), we can start at the origin O and proceed as follows:

    • First, move a units along the x-axis.

    • Then, move b units parallel to the y-axis.

    • Finally, move c units parallel to the z-axis.

Slide 25

3-D COORDINATE SYSTEMS

  • The point P(a, b, c) determines a rectangular box.

Slide 26

PROJECTIONS

  • If we drop a perpendicular from P to the xy-plane, we get a point Q with coordinates (a, b, 0).

    • This is called the projection of Pon the xy-plane.

Slide 27

PROJECTIONS

  • Similarly, R(0, b, c) and S(a, 0, c) are the projections of P on the yz-plane and xz-plane, respectively.

Slide 28

3-D COORDINATE SYSTEMS

  • As numerical illustrations, the points (–4, 3, –5) and (3, –2, –6) are plotted here.

Slide 29

3-D COORDINATE SYSTEMS

  • The Cartesian product R x R x R = {(x, y, z) | x, y, zR}

  • is the set of all ordered triples of real numbers and is denoted by R3.

Slide 30

3-D RECTANGULAR COORDINATE SYSTEM

  • We have given a one-to-one correspondence between points P in space and ordered triples (a, b, c) in R3.

    • It is called a 3-D rectangularcoordinatesystem.

Slide 31

3-D RECTANGULAR COORDINATE SYSTEM

  • Notice that, in terms of coordinates, the first octant can be described as the set of points whose coordinates are all positive.

Slide 32

2-D VS. 3-D ANALYTIC GEOMETRY

  • In 2-D analytic geometry, the graph of an equation involving x and y is a curve in R2.

  • In 3-D analytic geometry, an equation in x, y, and z represents a surfacein R3.

Slide 33

3-D COORDINATE SYSTEMS

Example 1

  • What surfaces in R3 are represented by the following equations?

  • z = 3

  • y = 5

Slide 34

3-D COORDINATE SYSTEMS

Example 1 a

  • The equation z = 3 represents the set {(x, y, z) | z = 3}.

    • This is the set of all points in R3 whose z-coordinate is 3.

Slide 35

3-D COORDINATE SYSTEMS

Example 1 a

  • This is the horizontal plane that is parallel to the xy-plane and three units above it.

Slide 36

3-D COORDINATE SYSTEMS

Example 1 b

  • The equation y = 5 represents the set of all points in R3 whose y-coordinate is 5.

Slide 37

3-D COORDINATE SYSTEMS

Example 1 b

  • This is the vertical plane that is parallel to the xz-plane and five units to the right of it.

Slide 38

NOTE

Note

  • When an equation is given, we must understand from the context whether it represents either:

    • A curve in R2

    • A surface in R3

Slide 39

NOTE

  • In Example 1, y = 5 represents a plane in R3.

Slide 40

NOTE

  • However, of course, y = 5 can also represent a line in R2 if we are dealing with two-dimensional analytic geometry.

Slide 41

NOTE

  • In general, if k is a constant, then

    • x = k represents a plane parallel to the yz-plane.

    • y = k is a plane parallel to the xz-plane.

    • z = k is a plane parallel to the xy-plane.

Slide 42

NOTE

  • In this earlier figure, the faces of the box are formed by:

    • The three coordinate planes x = 0 (yz-plane), y = 0 (xz-plane), and z = 0 (xy-plane)

    • The planes x = a, y = b, and z = c

Slide 43

3-D COORDINATE SYSTEMS

Example 2

  • Describe and sketch the surface in R3 represented by the equation y = x

Slide 44

3-D COORDINATE SYSTEMS

Example 2

  • The equation represents the set of all points in R3 whose x- and y-coordinates are equal, that is, {(x, x, z) | x R, z R}.

    • This is a vertical plane that intersects the xy-plane in the line y = x, z = 0.

Slide 45

3-D COORDINATE SYSTEMS

Example 2

  • The portion of this plane that lies in the first octant is sketched here.

Slide 46

3-D COORDINATE SYSTEMS

  • The familiar formula for the distance between two points in a plane is easily extended to the following 3-D formula.

Slide 47

DISTANCE FORMULA IN THREE DIMENSIONS

  • The distance |P1P2| between the points P1(x1,y1, z1) and P2(x2, y2, z2) is:

Slide 48

3-D COORDINATE SYSTEMS

  • To see why this formula is true, we construct a rectangular box as shown, where:

    • P1 and P2 are opposite vertices.

    • The faces of the box are parallel to the coordinate planes.

Slide 49

3-D COORDINATE SYSTEMS

  • If A(x2, y1, z1) and B(x2, y2, z1) are the vertices of the box, then

    • |P1A| = |x2 – x1|

    • |AB| = |y2 – y1|

    • |BP2| = |z2 – z1|

Slide 50

3-D COORDINATE SYSTEMS

  • Triangles P1BP2 and P1AB are right-angled.

  • So, two applications of the Pythagorean Theorem give:

    • |P1P2|2 = |P1B|2 + |BP2|2

    • |P1B|2 = |P1A|2 + |AB|2

Slide 51

3-D COORDINATE SYSTEMS

  • Combining those equations, we get:

  • |P1P2|2 = |P1A|2 + |AB|2 + |BP2|2

  • = |x2 – x1|2 + |y2 – y1|2 + |z2 – z1|2

  • = (x2 – x1)2 + (y2 – y1)2 + (z2 – z1)2

Slide 52

3-D COORDINATE SYSTEMS

  • Therefore,

Slide 53

3-D COORDINATE SYSTEMS

Example 3

  • The distance from the point P(2, –1, 7) to the point Q(1, –3, 5) is:

Slide 54

3-D COORDINATE SYSTEMS

Example 4

  • Find an equation of a sphere with radius r and center C(h, k, l).

Slide 55

3-D COORDINATE SYSTEMS

Example 4

  • By definition, a sphere is the set of all points P(x, y ,z) whose distance from Cis r.

Slide 56

3-D COORDINATE SYSTEMS

Example 4

  • Thus, P is on the sphere if and only if |PC| = r

    • Squaring both sides, we have |PC|2 = r2 or (x – h)2 + (y – k)2 + (z – l)2 = r2

Slide 57

3-D COORDINATE SYSTEMS

  • The result of Example 4 is worth remembering.

    • We write it as follows.

Slide 58

EQUATION OF A SPHERE

  • An equation of a sphere with center C(h, k, l) and radius r is: (x – h)2 + (y – k)2 + (z – l)2 = r2

    • In particular, if the center is the origin O, then an equation of the sphere is: x2 + y2 + z2 = r2

Slide 59

3-D COORDINATE SYSTEMS

Example 5

  • Show that

  • x2 + y2 + z2 + 4x – 6y + 2z + 6 = 0

  • is the equation of a sphere.

  • Also, find its center and radius.

Slide 60

3-D COORDINATE SYSTEMS

Example 5

  • We can rewrite the equation in the form of an equation of a sphere if we complete squares:

  • (x2 + 4x + 4) + (y2 – 6y + 9) + (z2 + 2z + 1)

  • = –6 + 4 + 9 + 1

  • (x + 2)2 + (y – 3)2 + (z + 1)2 = 8

Slide 61

3-D COORDINATE SYSTEMS

Example 5

  • Comparing this equation with the standard form, we see that it is the equation of a sphere with center (–2, 3, –1) and radius

Slide 62

3-D COORDINATE SYSTEMS

Example 6

  • What region in R3 is represented by the following inequalities?

  • 1 ≤ x2 + y2 + z2 ≤ 4

  • z ≤ 0

Slide 63

3-D COORDINATE SYSTEMS

Example 6

  • The inequalities

  • 1 ≤ x2 + y2 + z2 ≤ 4

  • can be rewritten as:

    • So, they represent the points (x, y, z) whose distance from the origin is at least 1 and at most 2.

Slide 64

3-D COORDINATE SYSTEMS

Example 6

  • However, we are also given that z≤ 0.

    • So, the points lie on or below the xy-plane.

Slide 65

3-D COORDINATE SYSTEMS

Example 6

  • Thus, the given inequalities represent the region that lies:

    • Between (or on) the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4

    • Beneath (or on) the xy-plane


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