P.1
PHYSICS
Vectors
1.1 Introduction
a) definition
scalar quantities can be completely specified by a magnitude. They can
be added algebraically.
vector quantities cab be specified by both a magnitude and a direction.
They can be added geometrically or analytically.
b)
symbols of vector quantities
Vectors quantities are usually represented by boldface type
e.g. V in printed material.

When written by hand, the designation V is often used.
c)
representation of vector
A vector is a line drawn to represent a vector quantity in
(i) direction, by its arrow head, and
(ii) magnitude, by its length.


e.g. consider 2 vectors A and B ,

B
we have
d)
example in mechanics
scalar quantities
length
l
mass
m
time
t
volume
V
speed
v
energy
W/E
pressure
P
density
q
temperature θ/T
potential
V
difference
(voltage)

A


A =-4B
vector quantities
displacement
force
velocity
acceleration
momentum
torque / couple
angular momentum

s

F

v

a

P
T
L
e)
The resultant and equilibrant
the resultant of a number of similar vectors is that single vector which
would have the same effect as all the original vectors taken together.
the equilibrant of a number of vectors is the single vector which is equal in
magnitude and oppositely directed to their resultant.
f)
Zero vector and until vectors
the null vector (or zero vector) is a vector with zero magnitude and no
specific direction.
-
a unit vector is a vector whose magnitude is 1.
P.2
  
The basic unit vectors in rectangular coordinates are i , j , k

i = unit vector in positive x direction

j = unit vector in positive y direction

k = unit vector in positive z direction
1.2 Addition of vectors
a) Parallelogram method (for adding 2 vectors)
the resultant of 2 vectors acting at any angle may be represented by the
diagonal of a parallelogram.
  
R  a b
-
b)

the resultant vector R can be measured graphically or calculated by cosine
rule.
Polygon method (adding any number of coplanar vectors geometrically)
Start at any convenient point and draw (in scale) each vector arrow in
turns. They may be taken in any order of succession. The tail end of
each arrow is attached to the tip end of the preceding one.
   
R  a b c
c)
Rectangular method (for both 2-dimensional and 3-dimensional cases)
Resolution and composition of vectors are usually resolved into vertical
and horizontal components. Then the resultant vector is the composite of
all the components together.
e.g. Find the resultant of the four coplanar forces act on a body at point 0
as shown below.
P.3
solution:
vector
80
100
110
160

R
+)
x-component
80
100 cos 45º
-110 cos 30º
-160 cos 20º
Rx = 94
y-component
0
100 sin 45º
110 sin 30º
-160 sin 20º
Ry = 71



Since R = Rx i + Ry j



i.e. R = -94 i + 71 j
R x2  R y2 =
R=
(94) 2  (71) 2 = 118N
tan θ= Ry / Rx
θ= -37ºor 143º (0º  θ  360º)
θ= 143º only (It is because R lies on the second quadrant)
e.g. Find the magnitude and direction of the vector d.
   
d  a b c
Given that



a  3i  4 j
 
b i


c 8j
solution:
vector

a

b

c

d
+)
hence,

d


=2 i +4 j
d
=
tanθ
θ
θ
d)
x-component
3
-1
0
dx = 2
d x2  d y2 =
= d y / dx =
22  42 =
y-component
-4
0
8
dy = 4
20
4
=2
2
= 63.4ºor 243.4º
= 13.4º only (it is because d lies on first quadrant)
Properties of vector addition
   
(i) a  b  b  a
(commutative law)
  
  
(ii) a  (b  c )  (a  b )  c (commutative law)
1.3 Subtraction of vectors
a) Parallelogram method

  
Since a  b  a  (b )
P.4
summary:
b)
Application on relative velocity

V A = velocity of A
If

V B = velocity of B

V AB = velocity of B w.r.t. A



then V AB = V B - V A
e.g. Two oars start at the same point with uniform velocities.
Car A moves to the north with speed 10m/s while car B moves to the east
with speed 4 ms-1. Find the velocity of car A observed by a passenger in
car B.
Solution:





V BA = V A = V B = -4 i + 10 j
VBA =
4 2  10 2 =
tanθ=
 10
4
θ
θ
= ________ or ________
= ________ only (it is because VBA lies on the second
quadrant)
e.g. Find the relative velocity of B w.r.t. A



V AB = V B – V A
VAB =
(4) 2  (8) 2  2(4)(8) cos 40
1.4 Multiplication of vectors
a)

Product of a scalar k and a vector a


K (a) = k a
P.5
-
It is defined as a new vector whose magnitude is k times the magnitude of

a . The new vector has the same direction as a if k is positive and
opposite in direction if k is negative.
b)


Scalar (dot) product of 2 vectors a and b
(i) definition – this product id a scalar and is equal to the product of the
magnitude of one vector and the component of the other
vector in the direction of the first.
(ii) properties
   
(1) a  b  b  a
      
(2) (a  b )  c  a  c  b  c
(iii) dot product of rectangular unit vectors
     
(1) i  i  j  j  k  k  1
     
(2) i  j  j  k  k  i  0
e)
Vector (cross) product of 2 vectors a and b
(i) definition – this product is a vector. Its magnitude is equal to the product
of the magnitude of one vector and the component of the other
vector in the direction perpendicular to the first.
 
axb  ab sin 
Its direction is perpendicular to the plane containing vectors
 
a , b and is determined by right hand screw rule.
(ii) Properties
 
 
(1) axb  b xa
      
(2) (a  b ) xc  axc  b xc
(iii) Cross product of rectangular unit vectors
    
(1) i xi  j xj  k xk  0
  
 
  
(2) i xj  k ; j xk  i ; k xi  j
P.6
d)
Examples of multiplication of vector qualities
(i) dot product e.g. work done on a body by a force F after a displacement
s.

 


W = F  s = F s cosθ (θis the angle between F and S )
frictionless
surface
W
frictionless
surface
= (10) (5) cos 0
= 50J
˚
W
= (10) (5) cos 60˚
= 25J
(ii) cross product
e.g. magnetic force acting on a current carrying wire,

B = magnetic field strength

I = current

1 = length of the wire (in the direction of the
current)

F = magnetic force induced on the wire



Since F = I 1 x B ,

F is directed perpendicularly into this page which is consistent with the
result obtained by using Fleming’s left hand rule.