16.11 Exercises 473
37.Show thata 1 z 1 bis orthogonal to aand b.
38.The quantitya 1
·
1 (b 1 z 1 c)is called a triple scalar product. Show that
(i)a 1
·
1 (b 1 z 1 c) 1 = 1 c 1
·
1 (a 1 z 1 b) 1 = 1 b 1
·
1 (c 1 z 1 a) (ii)
39.The quantitya 1 z 1 (b 1 z 1 c)is called a triple vector product.
(i)By expanding in terms of components, show thata 1 z 1 (b 1 z 1 c) 1 = 1 (a 1
·
1 c)b 1 − 1 (a 1
·
1 b)c
(ii)Confirm this formula for the vectorsa 1 = 1 (1, 3, −2),b 1 = 1 (0, 3, 1),c 1 = 1 (0, −1, 2).
40.Find the area of the parallelogram whose vertices (in the xy-plane) have coordinates
(1, 2), (4, 3), (8, 6), (5, 5).
41.The force Facts on a line through the point A. Find the moment of the force about the
point O for
(i)F 1 = 1 (1, −3, 0), A(2, 1, 0), O(0, 0, 0) (ii)F 1 = 1 (0, 1, −1), A(1, 1, 0), O(1, 0, 2)
(iii)F 1 = 1 (1, 0, −2), A(0, 0, 0), O(1, 0, 3)
42.Calculate the torque experienced by the system of chargesq
1
1 = 12 , q
2
1 = 1 − 3 andq
3
1 = 11 at
positionsr
1
1 = 1 (3, −2, 1),r
2
1 = 1 (0, 1, 2),andr
3
1 = 1 (0, 2, 1), respectively, in the electric field
E 1 = 1 −k.
43.A charge qmoving with velocity vin the presence of an electric field Eand a magnetic
field Bexperiences a total forceF 1 = 1 qE 1 + 1 qv 1 × 1 Bcalled the Lorentz force. Calculate the
force acting on the chargeq 1 = 13 moving with velocityv 1 = 1 (2, 3, 1)in the presence of the
electric fieldE 1 = 12 iand magnetic fieldB 1 = 13 j.
44.Use the property of the triple vector product (Exercise 39) to derive equation (16.60)
from equations (16.58) and (16.59).
45.The position of a particle of mass mmoving in a circle of radius Rabout the z-axis with
angular speed ωis given by the vector functionr(t) 1 = 1 x(t)i 1 + 1 y(t)j 1 + 1 zkwherex(t) 1 = 1 R 1 cos 1 ωt,
y(t) 1 = 1 R 1 sin 1 ωt,z 1 = 1 constant. (i)What is the angular velocity yabout the z-axis? (ii)Find
the velocity vof the particle in terms of ω, x, y, and z. (iii) Find the angular momentum of
the particle in terms of ω, x, y, and z. (iv) Confirm equation (16.60) in this case.
46.For the system in Exercise 45, show thatl 1 = 1 Iywhenz 1 = 10 , where Iis the moment of
inertia about the axis of rotation.
47.The total angular momentum of a system of particles is the sum of the angular momenta
of the individual particles. If the system in Exercise 45 is replaced by a system of two
particles of mass m with positionsr
1
1 = 1 x(t)i 1 + 1 y(t)j 1 + 1 zkandr
2
1 = 1 −x(t)i 1 − 1 y(t)j 1 + 1 zk, find
the total angular momentuml 1 = 1 l
1
1 + 1 l
2
, and show thatl 1 = 1 Iywhere Iis the total moment
of inertia about the axis of rotation. This example demonstrates thatl 1 = 1 Iywhen the axis
of rotation is an axis of symmetry of the system.
Section 16.8
Find the gradient∇ffor
48.f 1 = 12 x
2
1 + 13 y
2
1 − 1 z
2
49.f 1 = 1 xy 1 + 1 zx 1 + 1 yz 50.f 1 = 1 (x
2
1 + 1 y
2
1 + 1 z
2
)
− 122
Section 16.9
Find div 1 vand curl 1 vfor
51.v 1 = 1 xi 1 + 1 yj 1 + 1 zk 52.v 1 = 1 zi 1 + 1 xj 1 + 1 yk 53.v 1 = 1 yzi 1 + 1 zxj 1 + 1 xyk
54.Show that curlv 1 = 10 ifv 1 = 1 grad 1 f. 55.Show that div 1 v 1 = 10 ifv 1 = 1 curl 1 w.
abc
·
()z = ( )
aaa
bbb
ccc
x yzxyzxyzdeterminant