472 Chapter 16Vectors
Fora 1 = 1 (1, 2, 3),b 1 = 1 (−2, 3, −4),c 1 = 1 (0, 4, −1), find
6.a 1 + 1 b, b 1 + 1 a 7. 3 a, −a, a 2 38. 3 a 1 + 12 b 1 − 13 c 9. 3 a 1 − 13 c, 3(a 1 − 1 c)
10.|a 1 + 1 b|, |a| 1 + 1 |b|
11.Three masses,m
1
1 = 1 2, m
2
1 = 1 3, m
3
1 = 11 , have position vectorsr
1
1 = 1 (3, −2, 1), r
2
1 = 1 (2, −1, 0),
r
3
1 = 1 (0, 1, −2), respectively. Find (i)the position vector of the centre of mass,(ii)the
position vectors of the masses with respect to the centre of mass.
12.Three charges,q
1
1 = 1 3, q
2
1 = 1 −2, q
3
1 = 11 , have position vectorsr
1
1 = 1 (2, 2, 1), r
2
1 = 1 (2, −2, 3),
r
3
1 = 1 (0, −4, −3), respectively. Find (i)the dipole moment of the system of charges with respect
to the origin, (ii)the position of the point with respect to which the dipole moment is zero.
13.Forces are said to be in equilibrium if the total force is zero. Findfsuch thatf,
f
1
1 = 12 i 1 − 13 j 1 + 1 kandf
2
1 = 12 j 1 − 1 kare in equilibrium.
14.For the vectorsa 1 = 12 i 1 − 1 j 1 + 13 k, b 1 = 12 j 1 − 12 k, c 1 = 1 −k, find (i) x 1 = 12 a 1 + 1 b 1 + 14 c, (ii) a vector
perpendicular to cand x, (iii)a vector perpendicular to band c.
Section 16.4
Differentiate with respect to t.
- 2 ti 1 + 13 t
2
j 16.(cos 12 t, 3 1 sin 1 t, 2t)
17.A body of mass mmoves along the curver(t) 1 = 1 x(t)i 1 + 1 y(t)j, wherex 1 = 1 atandy 1 = 1 (at 1 − gt
2
)
at time t. (i)Find the velocity and acceleration at time t. (ii)Find the force acting on the body.
Describe the motion of the body (iii)in the x-direction, (iv)in they-direction, (v)overall.
18.A body of mass mmoves along the curver(t) 1 = 1 x(t)i 1 + 1 y(t)j 1 + 1 z(t)k, wherex 1 = 121 cos 13 t,
y 1 = 121 sin 13 tandz 1 = 13 tat time t. (i)Find the velocity and acceleration at time t. (ii)Find
the force acting on the body. Describe the motion of the body (iii)in the x- and
y-directions, (iv)in the xy-plane, (v)in thez-direction, (vi)overall.
Section 16.5
Fora 1 = 1 (1, 3, −2),b 1 = 1 (0, 3, 1),c 1 = 1 (1, −1, −3), find:
19.a 1
·1 b, b 1
·1 a 20.(a 1 − 1 b) 1
·1 c, a 1
·1 c 1 − 1 b 1
·1 c 21.(a 1
·1 c)b
22.Show thata 1 = 1 (1, 2, 3),b 1 = 1 (0, −3, 2)andc 1 = 1 (−13, 2, 3)are orthogonal vectors.
23.Find the value of λfor whicha 1 = 1 (λ, 3, 1)andb 1 = 1 (2, 1, −1)are orthogonal.
Fora 1 = 1 i,b 1 = 1 j,c 1 = 12 i 1 − 13 j 1 + 1 k, find
24.a 1
·1 b 25 .b 1
·1 c 26.a 1
·1 c
27.Find the angles between the direction of and the x-, y-, and z-directions.
28.A body undergoes the displacement dunder the influence of a forcef 1 = 13 i 1 + 12 j. Calculate
the work done (i) by the force on the body whend 1 = 12 i 1 − 1 j, (ii) by the body against the
force whend 1 = 1 i 1 − 13 k, (iii) by the body against the force whend 1 = 12 k.
29.A body undergoes a displacement fromr
1
1 = 1 (0, 0, 0)tor
2
1 = 1 (2, 3, 1)under the influence
of the conservative forceF 1 = 1 xi 1 + 12 yj 1 + 13 zk. (i)Calculate the workW(r
1
1 → 1 r
2
)done on
the body. (ii)Find the potential-energy functionV(r) of which the components of the
force are (–) the partial derivatives. (iii)Confirm thatW(r
1
1 → 1 r
2
) 1 = 1 V(r
1
) 1 − 1 V(r
2
).
30.Calculate the energy of interaction between the system of chargesq
1
1 = 12 ,q
2
1 = 1 − 3 and
q
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, and the
applied electric fieldE 1 = 1 − 2 k.
Section 16.6
Fora 1 = 1 (1, 3, −2),b 1 = 1 (0, 3, 1),c 1 = 1 (0, −1, 2), find:
31.a 1 z 1 b, b 1 z 1 a 32.b 1 z 1 c, |b 1 z 1 c| 33.(a 1 + 1 b) 1 z 1 c, a 1 z 1 c 1 + 1 b 1 z 1 c
34.a 1 z 1 c 1 + 1 c 1 z 1 a 35.(a 1 z 1 c) 1
·
1 b 36 .a 1 z 1 (b 1 z 1 c), (a 1 z 1 b) 1 z 1 c
aij k=− + 2