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1111
CHAPTER 3 VECTORS AND THE GEOMETRY OF SPACE
1
1 1= 7
torque vector is lr
x
Fl
=
)rlIF(sinO
where 6 is the angle between the position and force vectors. Observe that the only component of F that can cause a rotation is the one perpendicular to r, that is, I F sin 6. The magnitude of the torque is equal to the area of the parallelogram determined by r and F.
I
EXAMPLE 6 A bolt is tightened by applying a 40N force to a 0.25m wrench as shown
/
in Figure 5. Find the magnitude of the torque about the center of the bolt.
SOLUTION The magnitude of the torque vector is
/
/
1 T 1 = 1 r X F ( = ( r 1 1 F I sin 75" = (0.25)(40) sin 75"
=
10 sin 75"
. =
9.66 Nm
If the bolt is rightthreaded, then the torque vector itself is
I
FIGURE 5
where n is a unit vector directed down into the page.
17 Find the cross product a X b and verify that it is orthogonal to both a and b.
1415 Find 1 u X v I and determine whether u X v is directed into the page or out of the page.
2. a = ( I , I ,  I ) ,
3. a = i + 3 j  2k,
4. a = j + 7 k ,
b
=
(2,4,6)
b = i + 5 k
b=2ij+4k b=fi+j+fk b = 2 i + e'j e'k (1,2t, 3r2)
S.a=ijk,
6. a = i
+ e ' j + e'k,
(t, t2, t 3 ) , b
=
116.1 The figure shows a vector a in the xyplane and a vector b in
a
=
the direction of k. Their lengths are I a I = 3 and I b I = 2. (a) Find 1 a X b 1. (b) Use the righthand rule to decide whether the components of a X b are positive, negative, or 0.
8. If a = i  2 k and b
= j + k, find a X b. Sketch a, b, and a X b as vectors starting at the origin.
912 Find the vector, not with determinants, but by using properties of cross products. 9. ( i X j ) X k
11. ( j  k) X ( k  i )
lo. k X (i  2 j )
12. (i
+ j)
X (i  j)
State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. (b) a X (b c) (a) a (b X c) (c) a X (b X c) (d) (a b) X c (e) (a b) X (c d) ( f ) (a X b ) . (c X d)
17. I f a
.
=
( 1 , 2 , I ) and b = (0, l , 3 ) , f i n d a X b a n d b X a.

18. I f a = ( 3 , 1 , 2 ) , b = (1, l , O ) , a n d c = (0,0,4),show that a X (b X c) # (a x b) x c.
.
119.1 Find two unit vectors orthogonal to both ( I,  I , 1 ) and
(0,4,4).
SECTION 13.4 THE CROSS PRODUCT
1111
829
20. Find two unit vectors orthogonal to both i
and 2i
I
+ k.
+j + k
40. Find the magnitude of the torque about P if a 36lb force is applied as shown.
21. Show that 0 X a
=
0
b
=
a X 0 for any vector a in V 3 . 0 for all vectors a and b in V 3 .
22. Show that (a X b)
=
23. Prove Property I of Theorem 8.
24. Prove Property 2 of Theorem 8. 25. Prove Property 3 of Theorem 8. 26. Prove Property 4 of Theorem 8. 27. Find the area of the parallelogram with vertices A(2, I), B(O,4), C(4,2), and D(2,  I). 28. Find the area of the parallelogram with vertices K(l, 2,3),
L(1,3,6), M(3,8,6), and N(3,7,3).
2932 (a) Find a nonzero vector orthogonal to the plane through
41. A wrench 30 cm long lies along the positive yaxis and grips a bolt at the origin. A force is applied in the direction (0.3, 4) at the end of the wrench. Find the magnitude of the force needed to supply 100 N.m of torque to the bolt. 42. Let v = 5 j and let u be a vector with length 3 that starts at the origin and rotates in the xyplane. Find the maximum and minimum values of the length of the vector u X v. In what direction does u X v point?
the points P, Q, and R, and (b) find the area of triangle PQR.
m P(1,0,0),
30. P(2, P(0, 2, O),
Q(0,2,0), R(0,0,3) Q( 1,3,4), R(3,0,6) Q(4, 1, 21, R(5.3, 1) R(4,3,  1)
143.1 (a) Let P be a point not on the line L that passes through the
points Q and R. Show that the distance d from the point P to the line L is
I
32. P( 1,3, 11, Q(0,5,2),
3334 Find the volume of the parallelepiped determined by the vectors a, b, and c. 33.a=(6,3,I), 34.a=i+jk, b=(0,1,2), b=ij+k, c=(4,2,5) c=i+j+.k
+ + where a = QR and b = QP. (b) Use the formula in part (a) to find the distance from the point P(1, 1, 1) to the line through Q(0, 6y8) and R(1,4,7).
44. (a) Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is I(a x b) . c l d= laxbl + + + where a = QR, b = QS, and c = QP. (b) Use the formula in part (a) to find the distance from the point P(2, 1,4) to the plane through the points Q(l, 0, O), R(O,2, O), and S(0, 0, 3).
3536 Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. 35. P(2,0, 11,
36. P(3,0, l), 
Q(4, 1.01,

R(3, 1, l), $2, 2,2) S(0,4,2) 
Q( 1,2,5), R(5, 1, I),
1
Use the 5 .ar triiple u =: i  2 k, v are coplanar.
+
verify thalt the v and w = 5 i +
1
38. Use the scalar triple product to determine whether the points
145.1 Prove that (a  b) X (a + b) = 2(a X b).
46. Prove Property 6 of Theorem 8, that is, a X (b X c)
=
A(l,3,2), B(3, l,6), C(5,2, O), and D(3,6; 4) lie in the same plane. 39. A bicycle pedal is pushed by a foot with a 60N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P.
( a . c ) b  ( a . b)c
47. Use Exercise 46 to prove that
48. Prove that
Suppose that a # 0. (a) If a . b = a c, does it follow that b
=
c?
838
1111
CHAPTER I 3 VECTORS AND THE GEOMETRY OF SPACE
13.5
EXERCISES
16. (a) Find parametric equations for the line through (2,4,6) that
I. Determine whether each statement is true or false.
(a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) 'Tbo lines parallel to a plane are parallel. ( f ) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) l b o planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect are or parallel.
25 Find a vector equation and parametric equations for the line. 2. The line through the point (6, 5,2) and parallel to the
is perpendicular to the plane x planes?
y
+ 32 = 7.
(b) In what points does this line intersect the coordinate
17. Find a vector equation for the line segment from (2, to (4,6, 1).
l,4)
18. Find parametric equations for the line segment from (10,3, 1)
to (5,6, 3).
1922 Determine whether the lines Ll and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.
119./ L I : x = 6t,
y
=
1
+ 9t,
z
=
3t
vector (1,3,
f)
3. The line through the point (2,2.4,3.5) and parallel to the vector3i 2 j  k
+
4. The line through the point (0, 14,  10) and parallel to the line
x=1
+2t,y=63t,z=3+9t
22. L,:
The line through the point ( l , 0 , 6 ) and perpendicular to the plane x + 3y + z = 5
612 Find parametric equations and symmetric equations for the
=
x1 2
y3 2
22 1
line.
6. The line through the origin and the point (1,2,3)
7. The line through the points (1,3,2) and (4,3,0)
8. The line through the points (6, 1, 3) and (2,4,5) 2338 Find an equation of the plane. 23. The plane through the point (6, 3,2) and perpendicular to the
vector (2, 1 , s )
24. The plane through the point (4,0, 3) and with normal vector j + 2 k
The line through the points (0, f , 1) and (2, 1, 3)
10. The line through (2, 1,O) and perpendicular to both i and j k
+
+j
25. The plane through the point (I, 1, 1) and with normal vector i j  k
+
I I. The line through (1,  1, 1) and parallel to the line
x+2=fy=z3
12. The line of intersection of the planes x
26. The plane through the point (2, 8, 10) and perpendicular to the line x = 1 r, y = 2t, z = 4  3r
+y +z = 1
+
andx+ z
=0
27. The plane through the origin and parallel to the plane 2xy+3z=1
28. The plane through the point ( 1,6, 5) and parallel to the
113.1 Is the line through (4,
6, 1) and (2,0 3) parallel to the line through (10, 18,4) and (5,3, 14)?
planex+ y + z 3x7z= 12
+2 = 0
14. Is the line through (4, 1,  1) and (2,5,3) perpendicular to the
29. The plane through the point (4, 2.3) and parallel to the plane 30. The plane that contains the line x = 3
line through (3,2,0) and (5, 1,4)?
15. (a) Find symmetric equations for the line that passes through
and is parallel to the plane 2 x
the point (1, 5,6) and is parallel to the vector (1,2, 3). (b) Find the points in which the required line in part (a) intersects the coordinate planes.
+ 4y + 82 = 17
+ 2t, y = t, z = 8  t
~Theplanethroughthepoints(O,1,1),(1,0,1),and(1,1,0)
32. The plane through the origin and the points (2, 4.6)
and (5, 1,3)
i
SECTION 13.5 EQUATIONS OF LINES AND PLANES
1111
839
5758 Find symmetric equations for the line of intersection of the planes. 34. The plane that passes through the point (1,2, 3) and contains the line x = 3t, y = 1 + t, z = 2  t 35. The plane that passes through the point (6,0, 2) and contains the line x = 4  2t, y = 3 5t, z = 7 4 t 57. 5x  2y  22 = 1, 4x
+y +z =6
+
+
I
36. The plane that passes through the point (1,  1, 1) and contains the line with symmetric equations x = 2y = 32 37. The plane that passes through the point ( 1,2, 1) and contains the line of intersection of the planes x y  z = 2 and 2xy+3z=1
59. Find an equation for the plane consisting of all points that are equidistant from the points (1,0, 2) and (3,4,0). 60. Find an equation for the plane consisting of all points that are equidistant from the points (2,5, 5) and (6, 3, 1).
+
161.1 Find an equation of the plane with xintercept a, yintercept b,
and zintercept c. 62. (a) Find the point at which the given lines intersect:
38. The plane that passes through the line of intersection of the planes x  z = 1 and y + 2.7 = 3 and is perpendicular to the plane x y  22 = 1
+
3942 Use intercepts to help sketch the plane.
+ 5y + z = 10 41. 6x  3y + 42 = 6
39. 2x
40. 3x 42. 6x
+ y + 2.2 = 6
(b) Find an equation of the plane that contains these lines. 63. Find parametric equations for the line through the point (0, 1,2) that is parallel to the plane x y z = 2 and perpendicular to the line x = 1 t, y = 1  t, z = 2t.
+ 5y  32 = 15
+
++
4345 Find the point at which the line intersects the given plane. 43.x=3t, y = 2 + t , z=5t; xy+2z=9 x+2yz+l=O
64. Find parametric equations for the line through the point (0, 1, 2) that is perpendicular to the line x = 1 + t, y = 1  t, z = 2t and intersects this line. 65. Which of the following four planes are parallel? Are any of them identical?
44.x=1 +2t, y = 4 t , z = 2  3 t ; 45.x=y1=2z; 4xy+3z=8
46. Where does the line through (1,0, 1) and (4, 2.2) intersect the plane x + y + z = 6 ? 47. Find direction numbers for the line of intersection of the planes x + y + z = landx+z=O. 48. Find the cosine of the angle between the planes x a n d x + 2y + 3 z = 1. 66. Which of the following four lines are parallel? Are any of them identical? L,: x = l + t , y=t, z=25t
+y +z =0
4 : x + 1=y2=1z L3:x=I+t, L4: r
=
4954 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. m ~ + 4 ~  3I , ~ =~ x + ~ Y + ~ z = o 50. 22
= 4y
y=4+t,
z=1t
(2, 1, 3)
+ t(2,2, 10)
 x,
3x  12y + 62 xy+z=l x
=
1
6768 Use the formula in Exercise 43 in Section 13.4 to find the distance from the point to the given line.
51.x+y+z=l, 52. 2x  3y 53. x 54. x
= 4y
+ 42 = 5,
+ 6y + 4.2 = 3
 2.7,
8y = 1 2x
+ 2x + 42
 y + 22 = 1
6970 Find the distance from the point to the given plane. 69. (1, 2,4), 3x
+ 2y + 22 = 1,
+ 2y + 62 = 5
=
5556 (a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes. 55.x+y+z=l,
56. 3 x  2 y + z =
70. (6,3,5),
x  2y  42
8
x+2y+2z=1 1, 2 x + y  3 z = 3
7172 Find the distance between the given parallel planes. 71. 2 x  3 y + z = 4 , 4x6y+2z=3