CHAPTER-01

VECTORS
LEVEL 01 Starter Level
1.1Addition and Subtraction 9 Ifvectors P,Q and Rhave magnitudes 5, 12 and 13
1 Two vectors have magnitudes 3 unit and 4 unit, units and P+Q =R,then find the angle between
respectively. What should be the angle between (a) P and Q (b) Qand R (c) P and R
them if the magnitude of the resultant is 10 Five equal forces of 10 Neach are applied at one
(a) 1 unit (b) 5 unit (c) 7 unit point and allare lying in one plane. If the angles
between them are equal, then find the resultant of
2 Two forces, each of magnitude F have a resultant of
the same magnitude F.Find the angle between these forces.
them. 11 Which of the following sets of forces may be in
equilibrium?
3 The resultant of two forces 2p and 2p is V10p. Find (a) F, =3NE, =5N F = 1N
the angle between them.
(b) F, =3NE, =5NF, =9N
4 The square of the resultant of two forces 4N and 3N (c) F =3NE=5NF = 6N
exceeds the square of the resultant of the two forces (d) F=3NF, =5NF =15N
by 12 when they are mutually perpendicular. Find
the angle between the vectors. 12 The sum of magnitudes of two forces acting at a
point is 16N. If the resultant force is 8N and its
5 Two forces of 4 dyne and 3 dyne act upon abody. direction is perpendicular to smaller force, then find
Find the maximum and minimum value of the the forces.
resultant force on the body. 13 Two vectors Aand Bare such that A+B=A- B|.
6 A vector A makes an angle of 20° and B makes an Find the angle between the vectors A and B.
angle of 110° with the X-axis. The magnitudes of 14 The resultant of two vectors Aand Bis perpendicular
these vectors are 3m and 4m, respectively. Find their to the vector A and its magnitude is equal to half the
resultant. magnitude of vector B. Find the angle between A
7 Two forces each numerically equal to 10 dyne are and B.
acting as shown in the following figure, then find 15 The resultant of two opposite vectors of same
their resultant. magnitude is a .......
16 For the figure write the relation between the three
vectors.
10dyne C

60° 10 dyne A

make an angle of
17 Abody is moving uniformly on a circle with speed v.
8 Two vectors of 10 units and 5 units Find the magnitude of change in its velocity when it
120 with each other. Find the magnitude and angle has turned an angle 0.
of resultant with vector of 10 unit magnitude.
2 Problems in Physics

1.2 Vector Resolution 16 Atruck travelling towards North at 50 km/h, then

turns West and travels at the same speed. What is
11Î)+1| +lk| =..... the change in velocity?
2 A=2i +j,B =3 -k and C=6i -2k. Find value of 17 Aparticle moves from point P(2,3, 5) to point
A-2B +3C. Q(3, 4, 5). Find its displacement vector.
3 Ifa unit vector is represented by (0.5i -0.8j+ ck), 18 Abird moves from point (1,-2,3) to (4, 2, 3). If the
then find the value of c. speed of the bird is 10m/s, then find the velocity
vector of the bird.
4Forces 3N, 4N and 12N act at a point along X, Yand
19 A person moves 30m North, then 20 m East and then
Z-axis, respectively. Find the magnitude of the 30/2 mSouth-West. Find hisdisplacement from the
resultant force. original position.
5 Aforce F=6f-8j +10k Nproduces an acceleration 20 Asail boat sails 2 km towards East, 5 km 37°South of
of 5 ms in a body. Find the mass of body. East and finally an unknown displacement. If the final
displacement of boat from the starting point is 6 km
6 Find the angle that the vector A=2i+3j makes with
towards East, find the third displacement.
(a) X-axis (b)Y-axis (c) Z-axis.
7 IfA=3i +4j and B=7i +24j, find the vector having 21 Aparticle moving with initial velocity 2i +3, 4k
the same magnitude as Band parallel to A. has acceleration -i +2j +5k. Find its velocity at the
end of3 s.
8fA=2i+4j -sk,then find direction cosines of the
vector A. 22 Three forces P, Qand Rare acting at apoint in the
9 Ifavector Amakes angles a, Band Ywith the X, Y plane. The angle between P and Qand Qand Rare
and Z-axes respectively, then 150° and 120°, respectively. Then for equilibrium,
find ratio of forces P,Q and R.
(a) cosa+ cosB+ cos y =..
23 Three forces of 3N, 2N and 1N act on a particle as
(b) sin'a +sinB+ siny=.
shown in the figure. Calculate the
10 Vector A makes equal angles with X, Yand Z-axes.
2N
Find value of its components.
11 Find th¹ vector that, must be added to the vector
-6i 9j +2k and 3i +6j -7k, so that the resultant 60° 3N
vector isa unit vector along the Y-axis. 60°

12 Find the unit vector parallel to the resultant of the N

vectors A=4i +3j +6k and B= -i +3j -8k.
a) net force along the X-axis.
13 The x and y components of vector Aare 4 m and 6m, (b)net force along the Y-axis.
respectively. The x and y components of vector A+ B (c) single additional force required to keep the body in
are 10 mand 9 m, respectively. Calculate for the equilibrium.
vector Bthe following
(a) its x and y components, (b) its length and 24 If the four forces as shown in figure are in equilibrium,
then express F, and F, in unit vector form.
(c) the angle it makes with X-axis.
it Fz
14 A
particle is acted upon by the forces
F=2i+oj -sk,F, =si +j-bk. F, =bi +sj-7Â, 15 N
K
30) 10N
F, =ci +6j -ak.
Find the values of the constants a,b, c in order that 37°30°
the particle willbe in equilibrium.
15 Add vectors A,Band Ceach having magnitude of 100
unit and inclined to the X-axis at angle 45, 135° and
315°, respectively.
Vectors
3

- 4k.
1.3Dot Product and Cross Product 11 Given two vectors A=2i +2 -k and B=3i
Find
1 Write the values of the following. (a) A-B
(0j-k (ii) k-k (ii) a-å (b) angle between the two vectors
(iv) k-(q x) (c)projection of Aon B
(d) projection of Bon A.
2 Answer the following.
(0) Can 2similar vectors of different magnitude yield a 12 Find the work done by force F=i+2j-k, when a
zero resultant? Can 3 yield? body is dispacedfromr, =itjtor, =2i -3j +k.
(iü) Can a +b= a-b?
(ii) Time has a magnitude and direction. Is it a vector? 13 The torque of a force Fabout a point of position
vector r is given by t=rXF.Use this formula to find
(iv) When will axb=a-b?
torque of force 2i +5j about a point 2i +j-k.
(v) Does the unit vectors i, j and k have units?
14 Find a vector of magnitude 9, which is perpendicular
3 Find the scalar product of vectors A =2i+5k and to both the vectors 4i +j+3k and 2i +j+2k.
B=3+sk.
15 Find the area of the parallelogram whose sides are
4Find the angle between the two vectors i+2j +3k and 2i+j.
(02i +2j -kand j-k (i) 2i +j-kandí -k 16 Find the area of the parallelogram whose diagonals
() i +j+kand i (iv) 2i -2j -kand i +j are given by i +2j -kand i +j+2k.
() i+2j -kand -i+j-2k 17 a=5i +4j-6k, b=-2i +2j +3k and
5IfA =si+7) 3k and B=2i +2j - ck are c=4i+3j +2k. Find
perpendicular vectors, then find the value of c. (a) a x(b + ) (b) a x(b xc)
6 The scalar product of two vectors is 2/3 and 18 Given that a=î+j +k, b=-i +-k and
magnitude of their vector product is 2. Find the angle c=i+j-k, then evaluate
between them. () (a-b) + (b-c) + (c-a) (ii) (a-c)c +(c-b)a
7 IfP.Q=PQ, then find the angle between P and Q. 19 Prove that (a +b) x (a-b) = -2(axb)
8IF|AxB|= /3A-B, then find the value of |A +B|. 20 a=i+j. b=j+k. Then,
() using cross product find the angle between vectors
9 Find the value ofp for which the vectors aand b.
A=3i +2j +9k and B=i+pj +3k are (iü) Find area of the APOS formed by vectors a and b.
()perpendicular (ii) parallel.
21 (a) If A
points vertically upward and Bpoints towards
10 There are two vectors A=3i +j andB=j+2k. For East, then what is the direction of (AX B)?
these two vectors, Find (b) IfA is vertically downwards and Bis along South,
(0) the component of A along Band
(i) the component of A perpendicular to Bin vector then what is the direction of (A xB)?
form.

LEVEL 02 JEE Main Level

Single Choice Correct Questions
1Choose the correct statement. 2 Avector may change, if
(a)Electric current is a vector because it has both (a) frame of reference is translated
magnitude and direction. (b) frame of reference is rotated
(c) vector is translated parallel to itself
(b) Time is avector which has direction always in the (d) vector is rotated
forward direction.
(c) Allquantities having magnitude and direction are 3At what angle must the two forces (x +y) and (x -y)
vector quantities.
(d) None of the above
act, so that the resultant may be /x' +y')?
Level 1 1.2 Vector Resolution
1.1Addition andSubtraction 1. 3 2. 20i -5j- 4k 3. Jo.11 4. 13 N 5.2V2 kg
1. (a) 180° (b) 90° (c) 0º 2. 120° 3. 45°
4. 60° 5.7 dyne, 1 dyne
6.
(a) tan (b) tan (c) 90° 7. 15i +20j
6. 5 m 7. 10dyne 8. 30° 2 4 -5 A
8. 9.2 10.
V45' V45 V45 V3
9. (a) 90°

11. (c)
)
(b) cos (c) cos

12. 6 N and 10 N 13. 90°

) 10. Zero

14. 150° 11. 3i + 4j+ 5k 12.

3i+6j 2k
7
15. Null vector 16. A+C=B
17.2vsin 13. (a) 6 and 3 (b) 45 (c) tan
14. C=-4 15. 100 at an angle of 45° with X-axis
16. 50/2 km/h (South-West)
17. i+j 18.6i +8j 19. 10 m (West) 20.3 km (North)
21. -i+9j +11k 22. V3:2:1
23.(a) 1.5 N (b) V23 N()--
2 2
3;
24. F, =-(12/3 -1)j and F, =(12 -5/3)i +(12/3-15))
1.3Dot Product and Cross Product
1. () 0 (ii) 1 (iii) 1 (iv) 1 (v) 0
2. (i) No, Yes (ii) Yes, if b is a null vector (iii) No
(iv) Never (v) No
3. 25
30 ()
4. (i) 45° () 30 cos
(i) cos (iv) 90° (v) 60

5. -8 6. 30° and 150° 7.0°

2
8. (A?+B² +AB)1/2 9. (i) -15 (ii)
3

10. (0ú +2k) 3i +-*

11. (a) 10 (b)

12, -8J
13. 5Í -2j +8k 14. t(-31 -6j+6k)
16.
/35 Sq. units
15. /54 sq. units 2

17. (a) 50i-37j+17k (b) 40i +100j+ 100k

18. () 1 () 2í +25 19. - 2(a xb)

20. () 60 (1) sq. units 21. (a) North (b) West

2