Chapter 12.8 - Vectors

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  • 8/12/2019 Chapter 12.8 - Vectors

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    E S SON

    ~,', 2' ' 8,

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    The

    eometry

    Systems

    [Linearq.uationsinThree

    T1~ bl,rza es

    In Lesson 12-7, you saw that a linear equation

    ax + by + cz = d,

    in which at least one of the coefficients a, b, and c is nonzero, is the

    equation of a plane Min 3-space, and that the vector

    -7

    v = (a, b, c)

    is perpendicular to M. Consequently, the solution set to a system of klinear

    equations in three unknowns

    a1 x + b1 y + clz = dla2x + b 2 Y + c2z = d2

    akx + bky + ckz = dk

    has a geometric interpretation; it is the set of points common to the k

    planes defined by the k equations in the system.

    Finding the Intersection of Two Planes

    When there are two planes, that is, when k = 2, then there are threepossible situations for the location of the planes: they intersect in a line, or

    they are parallel and not coincident, or they coincide.For instance, consider the following system of two equations in

    three unknowns.

    {

    2X + 5y - Z=6x - 8y + 4z = 7

    Its solution i~the intersection of the two planes. Notice that plane (1) is

    perpendicular to the vector (2,5,-1) and plane (2) is perpendicular to the

    vector (1, -8, 4). Since these vectors are not parallel, the planes are neither

    parallel nor do they coincide. So the planes intersect in aline. To describe

    this line, solve for two of the variables in terms of the third variable. Here

    we solve for x and z in terms of y.

    (1)

    (2)

    2x - z = 6 - 5y (1)

    x + 4z = 7 + 8y (2)

    To solve this system for x, we multiply equation (1) by 4 and add the result

    to equation (2).

    So,

    8x - 4z = 24 - 20y

    x + 4z = 7 + 8y9x= 31 -12y

    31 - 121 'x = 9'

    4 . (1)

    (2)

    Lesson 12-8 The Geometry ofSystems of Linear Eauations in Three Variables 767

    - - - - - - - - - - - - - - - - ~ - - - - ~ ~ --

    I

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    and

    QIX + bly + CIZ = dlQ2X + b2y + C2Z = d2,

    Now repeat the process to solve for z. This time we multiply equation (2)

    by -2 and add the result to equation (1).

    -2x - 8z = -14 - 16y

    2x - z = 6 - 5y

    -9z = -8 - 21y8 + 21 v

    Z = 9'

    (31 - 121 ' 8 + 211')

    (x, y, Z ) = --9-" y, 9"

    So,

    Thus

    Because all three components depend ony , y is a parameter in this situation.

    To write the parametric equations with the usual variable t,sety =t.Then

    parametric equations for the line are as given below.

    (

    31 - 12tx = 9

    )' = t8 + 21t

    Z = 9

    To find a point on the line, pick a value of t.For example, t= 0 gives the

    point ( 3 91, 0, ~), while t= 2 gives (~, 2,5 9 ) . These two points satisfy the

    equations for both planes. This checks that the above parametric equations

    are for the line of intersection of the planes.

    If for two planes.

    there is a constant k such that

    k(al, bl, cI) = (a2, b2, c2),

    then the vectors perpendicular to these planes are parallel. Thus the planesare distinct and parallel, or they coincide. Ifkdl = d2, then the equationsare equivalent and so the planes coincide. Ifkdl * d2, then there is no pointIncommon.

    Finding the Points Common to Three Planes

    Now consider a system (*) of 3 linear equations in 3 unknowns.

    f

    alx + bl)' + clz = d1(*) a2x + b2y + c2z : d2

    Q3X

    + b

    3y + c

    3z - d

    3

    768

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    You are asked to explore the situation where the system (*) has infinitely

    many solutions in Question 6.

    QUESTIONS

    / Covering the Reading

    1. a. Solve the equations for planes (1) and (2) on page 767 for xand yin

    terms ofz .h. Show that the line of intersection is the same line as found on page 768.

    2. a. Sketch the planes corresponding to the two equations in the system

    {

    X + Y + z = 1y - z = O .

    h. Give parametric equations for the line of intersection of the planes.

    3. Consider the plane M with equation 3x - 2y + 4z =1.a. Write an equation for a plane N that is parallel to (but not coincident

    with) M. )

    h. Solve the system consisting of the eq~ations for M and N to show that

    there is no solution.

    In 4-5, describe all solutions to the system.

    4. { 5 X - 3y + z =12 5. { - x + 3y, - 2z =-6x + y - 4z = 1 2x - 6y + 4z =12

    6. When the system (*) in this lesson has infinitely many solutions, draw all

    possible configurations of the three planes MI, M2, and M3

    7. Consider the following system of equations given in this lesson:

    {

    X + Y + 3 z = 1x + Y + 3 z =2

    2x-y+ z=8.

    a. Which pair of equations corresponds to the parallel planes? How do

    you know?

    h. How can you tell that the other equation corresponds to a plane that

    intersects the parallel planes?

    8. Consider the following system given in this lesson:

    {

    X + 2y + z = 4

    2x- y+ z= 33x + Y+ 2z =-2.

    a. Show that the system has no solution.

    h. Choose one pair of the equations, and show that their graphs intersect

    in a line.

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    In 9-11, describe the graph of all solutions to the system.

    {

    a+ b+c=1

    9. 4a + 2b + c =39a + 3b + c =6

    {

    5u - 3v = w

    10. IO u - 2w = 6v

    u + v = 9

    {

    x+ y-z= 2

    11. x + 3y - z = 16-x + y + z =12, ~

    12. The points (1, 1),(2, 3), and (3, 6) are on the graph ofy = ax2 + bx + c.Find a, b,and c. '

    13. To get from the O'Neill farm to the city, you must travel d miles over a

    dirt road, g miles over a gravel road, and hmiles over a highway. When

    Mr. O'Neill averages 20 mph on dirt, 15 mph on gravel, and 60 mph on

    the highway, it takes him 68 minutes to get to the city. Mrs. O'Neill

    averages 5 mph slower than Mr. O'Neill on dirt and gravel, but averages60 mph on the highway, and it takes her 82 minutes. The total distance is

    42 miles. How many of these miles are over each surface?

    Review

    14. Describe the graph of the set of points (x, y, z) such that y + Z = 10.(Lesson 12-7)

    15. Find parametric equations for the linee passing through (-7, 2, 0) andperpendicular to the plane with equation 3x - 4y - z =-1. (Lesson 12-7)

    16. a. Find an equation for the plane containing the point (-7, 2, 0) which is

    p~rpendicular to linee given in Question 15.h. How is this plane related to the plane in Question 15? (Lesson 12-7)

    17. Explain why three nonzero 3-dimensional vectors U ; ,u ; " andU ;with thesame initial point are coplanar if and only ifU ; . c u ; , XU ; ) =O. (Lesson 12-7)

    18. Suppose g is a polynomial function such that

    g'(x)

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    19. Is the folIowing argument valid? Explain.

    If school isnor closed, then I go to school.

    I don't go to school.

    School is closed. (Lessons 1-6, 1-7)

    Exploration

    20. a. Find an equation for the plane containing the lines f 1 and f2 with the

    following parametric equations.

    {

    X =1 + 4t {X =1 - 2tf1: Y = 2 - 3t f2: Y = 2 + t

    Z = 3 + t Z = 3 + 4t

    h. Generalize part a to find an equation for the plane containing the lines

    with the following parametric equations.

    {

    X = Xo + u 1 t {X = X o + V I tfI: Y =Yo + u2t f2: Y =Y o + v2t

    Z =Zo + u3t Z =Zo + V3t

    772