Chapter 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines

 1.  For each of the following, determine whether the line intersects the plane, and if so how. 

a). line:     r = (1,5,1) + t(3, -9, 1)

     plane: 2x + y + 15z + 4 = 0

                     Answer: intersect at the point (2, 2, -2/3)

                                                             Video solution to 1a)


b). line:      r = (-2, 1, 8) + t(6, -3, -2)

      plane:   2x + 2y + 3z - 11 = 0

                        Answer: no intersection

                                                             Video solution to 1 b)


C). Line:        r = (0, -5, -9) + t(4, 1, 3)

      Plane:      x + 8y - 4z = 0

                         Answer: line lies on plane

                                                                Video solution to 1c)


 2. Determine whether each of the following pairs of lines intersects and if so, how they intersect. 

a) line: r = (2, -5, -7) + s(1, -9, 3)

    line: r = (0, 13, 1) + t(-2, 18, -6)

                            Answer: lines are coincident

                                                      Video solution to 2a)


b) line: r = (-3, 9, -4) + s(6, 3, 5)

    line: r = (-2, -8, 3) + t(18, 9, 15)

                            Answer: lines are parallel, non coincident

                                                      Video solution to 2b)


c) line: r = (2, 3, -9) + s(3, 4, -1)

    line: r = (-11, 11, 22) + t(1, -5, -7)

                            Answer: lines intersect at only one point (-7, -9, -6)

                                                      Video solution to 2c)

 

D) line: r = (-3, 1, 7) + s(1, 2, -5)

    line: r = (3, 9, 8) + t(5, 6, -3)

                            Answer: lines are skew

                                                      Video solution to 2d)


 3.  Determine the point at which the line r = (-3, 8, 4) + t (1, -2, 3) intersects the xz-plane 

                     Answer : (1, 0, 16)

                                 Video solution to #3


4.  Determine whether each of the following lines has a y-intercept

a) r = (10, 7, -15) + t(2, 4, -3)

                       Answer: yes (0, -27, 0)

                                        Video solution to 4a)

B). r = (-7, 8, 3) + t(9, -1, -8)

                          Answer: no

                                         Video solution to 4b)


 

 

Chapter 9.2 - Systems of Equations (and Appendix work with Matrices)

5. Write a solution to each of the following using parameters

a) 2x + 3y = 7
                           Answer: x = -3/2 t + 7/2
                                          y = t
                                                             Video solution to 5a)

b) 9x - 4y + 2z = 11
                             Answer: x = s
                                            y = t
                                             z = -9/2 s + 2t + 11
                                                              Video solution to 5b)

C)     4x + 3y - 9z = 12
         7x - y + 5z = -8
                              Answer: x = t
                                            y = -83/6 t - 2
                                            z = -25/6 t - 2
                                                                Video solution to 5 c)

6. Determine the values of k so that the following system of equations 

                 (k - 3)^2    x         +          4 y     =      84
                             4   x         +             y     =      3k

    a)  an infinite number of solutions
    b)  zero solutions
    c)  one solution

                                             Answer: a) k = 7
                                                            b) k = -1
                                                            c) k does not equal 7 or -1, and k is a real #

                           Video solution # 6

 

7.

Solve each of the following systems of equations using matrices and interpret the results:

a)

        2x - 2y + 3z = 14

        x + 4y - 2z = 11

        4x - 4y + 6z = 28

                        Answer:  No solution, inconsistent system.      Video solution to 7a)

b)

        3x + y - 4z = -5

        2x + 9y - 8z = 21

        4x - 2y + 5z = -41

                        Answer:  intersect at the point (-6, 1, -3).           Video solution to 7b)

c)

        2x - y + z = 3

        4x - 2y + 2z = 6

        x + y + z = 2

                        Answer:  intersect at a line.                                 Video solution to 7c)

d)

        3x + 2y + z = 12

        6xd + 4y + 2z = 24

        9x + 6y + 3z = 36

                        Answer:  3 coincident planes.                               Video solution to 7d)


 8.   Solve the following system of equations

          1/x    +    1/y     -    3/z    =    -16

          1/x    +    3/y    +    1/z    =     -8

          2/x    +    1/y    +    1/z    =     -9

                                              Ans: ( -1/5 , -1/2 , 1/3 )

                                                                                        Video solution to # 8

 

9.

For what value(s) of k will the following system of equations have

a)  an infinite number of solutions

b)  one solution

c)  zero solutions

                  2x + y + 4z = 3

                  20x - 5y - 128z = 10

                  5x + ky - z = 5

Answer:  a) k = 5/8;  b) impossible;  c) k does not equal 5/8, kER.                 Video solution to # 9      

  

Chapter 9.3 - The Intersection of Two Planes

 

10.

Determine whether each of the following system of equations is consistent or inconsistent.  If the system is consistent, then state the equation of the line of intersection or show that the planes are coincident

a)

        2x + 3y - 4z = 7

        6x + 9y - 12z = 22

                 Answer: inconsistent system; two parallel, non-coincident planes.             Video solution to 10a

b)

        3x + y + 8z = 23

        6x + 2y + 16z = 46

                 |Answer: consistent system; two coincident planes.                           Video solution to 10b

c)

        x - 4y + 3z = 4

        2x + y - z = 6

                 Answer:  line of intersection r = (22/7, 0, 2/7) + t(1,7,9), tER.                       Video solution to 10c

       

 11.  Show that the line r = (3, -1, 2) + t(4, -1, 3)  lies on the plane 5x-28y+16z-75=0

                            Video solution to #11


12. State the equation of the line 3x = 2y + 10 = -4z in vector form using only integers. 

                     Answer: r = (0, -5, 0) + t (4, 6, -3)

                                                                                       Video solution to #12


  

Chapter 9.4 - The Intersection of Three Planes

VIDEO  LESSON


When dealing with three planes,  there are 8 possible scenarios.  Here is a video lesson in two parts. 

                                              Part 1.                     Part 2. 


13.

Determine whether the following system of equations is consistent or inconsistent.  If the system is consistent, then state the single point of intersection or the equation of the line of intersection or show that the planes are coincident.

a)

        3x - y + 4z + 8 = 0

        2x + y + z + 7 = 0

        2x + 5y - 2z + 9 = 0

                 Answer:  consistent; single point of intersection (-5,1,2).     Video solution to 13a

b)

        x + y + z = 2

        4x - y - z = -7

        2x - y - z = -5

                 Answer:  consistent; line of intersection r = (-1,3,0) + t (0,1,-1), t ER.        Video solution to 13b

c)

        x + y = 3

        y + 2z = 7

        8x + 3y - 10z = 62

                 Answer: inconsistent system; triangular prism.             Video solution to 13 c

d)

       x + 4y - 5z = 7

       2x + 8y -10z = 17

       3x + 2y - 6z = 1

                 Answer:  inconsistent: 2 parallel non-coincident planes and a non-parallel plane intersecting both

       

                                      Video solution to 13 d


e)

        

             x + y - z = 4

         2x - y + 5z = 11

        6x - 3y + 15z = 33

                 Answer: consistent; line of intersection is r = (5, -1, 0) + t (4, -7, -3), t ER


                                             Video solution to 13 e


f)

        x - 2y + z = 4

        3x - 6y + 3z = 12

        x/2 - y + z/2 = 2

                Answer:  consistent; 3 coincident planes.                              Video solution to 13 f

g)

        4x - 3y + z = 10

        8x - 6y + 2z = 20

        x - 3/4 y + 1/4 z = 3

                 Answer:   inconsistent; 2 coincident planes and a third parallel, non-coincident plane


                                                     Video solution to 13 g


h)

        4x + 2y - z = 11

        8x + 4y - 2z = 23

        12x + 6y - 3z = 34

               Answer: inconsistent; 3 parallel, non coincident planes.                 Video solution to 13 h

 

14.

Determine a and b such that the following three planes

        3x + 2y + 5z = 10

        6x + y + 8z = a

        bx + 3y + 6z = 2

a) intersect along a line

b) do not intersect

c)  intersect at just one point

Answer: a) a = 158/3, b = 36/11;  b) b = 36/11, a does not equal 158/3, a ER;  

c)  b does not equal 36/11, a ER, b ER


                                                                              Video solution to 14



15.   Solve the following system of equations

                   4/x    -    2/y    +    10/z    =    3

                   3/x    +    4/y    -     3/z     =    3

                   2/x    -     6/y    +    2/z     =    5

                                                                             answer : (3/4, -2, -3)


                                                                      Video solution to 15




  

Chapter 9.5 - The Distance from a Point to a Line in R2 and R3


16. Calculate the distance between the following pairs of lines


a)   Line 1: r = (-9, 1) + t (4, -7)

      Line 2: r = (3, 8) + s (4, -7)

                                     Answer: 112 sqrt(65) / 65

                                                                      Video solution to 16a


b)  Line 1: 3x - 8y + 27 = 0

     Line 2: 3x - 8y - 9 = 0

                                       Answer: 36 sqrt(73) / 73

                                                                        Video solution to 16b


17.  Determine the distance from the point P to the line L in each of the following situations

a)  P(5, -1); L: 9x - 7y + 53 = 0

                                        Answer: 21 sqrt(130) / 26

                                                                         Video solution to 17a


b)  P(7, 6); L: r = (7, 6) + t (8, -1)

                                        Answer: 23 sqrt(5) / 4

                                                                         Video solution to 17b


18.  Determine the distance from the point (-7, -1, 5) to the line 

r = (-8, 2, -1) + t(11, -4, 1)

                                                 Answer: approximately 6.3

                                                                           Video solution to 18


19.  Determine the distance between the lines 

line 1: r = (3, -8, 2) + t(1, 7, 6)

line 2: r = (9, 7, 0) + s(1, 7, 6)

                                        Answer: approximately 12.3

                                                                              Video solution to 19


20. Blank


21.  Determine the coordinates of the point on the line r = (48, -18, 14) + t(6, -9, 7) that produces the shortest distance between the line and a point with coordinates (-6, -8, 2)

                              Answer: (30, 9, -7)

                                                                   Video solution to 21


22.  The point P(5, -3, 6) is reflected in the line with equation 

r = (2, -1, 4) + t(3, -1, -2) to give the point P'. Determine the coordinates of P'  

                              Answer: (2, 0, 0)

                                                                   Video solution to 22

 

  

Chapter 9.6 - The Distance from a Point to a Plane


23.  Determine the distance from the point P(-7, 4, -5) to the plane 

2x - 3y + 8z - 76 = 0

                                      Answer: 142 sqrt(77) / 77

                                                                            Video solution to 23


24. The distance from the point (1, -2, 1) to the plane 2x + y + az - 4 = 0 is 2 units. Solve for a. 

                         Answer: a = -2/3 or a = -2

                                                                            Video solution to 24


25. Determine the distance between the lines 

    line 1: r = (-3, 4, 0) + t(1, -6, 3)      and      line 2: r = (2, -1, -5) + s(3, 8, 9)

                          Answer: 2 sqrt(10)

                                                                            Video solution to 25


26. Determine the coordinates of points on line 1 and line 2 that produce the minimal distance between the lines

    line 1: r = (1, 2, -1) + t(1, 1, 2)      and      line 2: r = (-4, 4, -5) + s(1, -1, 1)

                          Answer: A (3/14, 17/14, -18/7)   and   B (-6/7, 6/7, -13/7)

                                                                           Video solution to 26


  

Chapter 9 Review


27.

Show that the points A(5, 10, -40), B(0, 60, 0), C(-30,6,4) and D(-15, 15, -10) all lie on the same plane.

                                                                                              Video solution to 27


28.

A perpendicular line is drawn from the point A(26, -1, -2) to the plane 4x-8y+3z+72=0 and meets the plane at point B.  Determine the coordinates of point B

Answer: (18, 15, -8)

                                                             Video solution to 28


29.

Point (-2, -8, 6) is reflected in the plane with equation 3x+7y-z-50=0.  Determine the coordinates of the image point.

Answer:(10, 20, 2)

                                                                Video solution to 29