MOTION IN A PLANE

VECTORS
SCALARS AND VECTORS

All the physical quantities can be divided into two types.

They are:
(i) Scalars and
(ii) Vectors

The basic difference between them is the ′direction′.

Scalar
A physical quantity which has a magnitude but no direction is called scalar.

Example: Mass, length, temperature, time, density, work, specific heat, etc.

A scalar may be positive or negative. Scalars can be added, subtracted,

multiplied or divided according to rules of ordinary algebra.

Addition and subtraction of scalars make sense only for quantities of same
nature.
However, multiplication and division can be performed with different
quantities.

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Vector
A physical quantity which has both magnitude and direction is called vector.
Example: Velocity, acceleration, displacement, force, momentum, torque, etc.

Vectors cannot be added, subtracted, multiplied or divided according to simple

rules of algebra.

Vectors can be added, subtracted or multiplied according to the rules of vector

algebra.

A vector cannot be divided by another vector as it is not a valid operation in

vector algebra.

A vector quantity obeys the ′triangle law of addition′ or equivalently the

′parallelogram law of addition′.
A vector is represented as a bold face type.
For example, velocity vector is represented as v.
While writing it is difficult to reproduce bold face type. Therefore, a vector can
also be represented as v.
Magnitude (scalar part) of a vector is represented as a light face type.
For example, magnitude of velocity vector is represented as v or | v |.

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Equality of Vectors
P S
Two vectors A and B are said to be equal if,
and only if, they have the same magnitude A B
and the same direction.
The figure shows two equal vectors A and B.
To check their equality, shift B parallel to itself O Q
until its tail Q coincides with that of A,
i.e. Q coincides with O.
P′
Since their heads S and P also coincide,
the two vectors are said to be equal. A′
In general, equality is indicated as A = B.
S′
Vectors A′ and B′ have the same B′
magnitude but they are not equal O′
Q′
because they have different directions.
Even if we shift B′ parallel to itself so
that its tail Q′ coincides with the tail O′ A″
O″ P″
of A′, the tip S′ of B′ does not coincide
with the tip P′ of A′. B″
Q″ S″
Vectors A″ and B″ have the same
direction but they are not equal because
they have different magnitudes. Home Next Previous
Position and Displacement Vectors Y A
To describe the motion of an object in a
plane, its position with respect to origin can r′ - r P′
be chosen conveniently. P
r
Let P be the position of the object in X-Y r′
plane at time t. B

Draw a straight line from origin O to P and O X

place an arrow head at P on the line OP.
The point O of OP is called ′tail′ and the point P of OP is called ′head′ of
vector OP .
OP is called ′position vector′ of the object. It is represented by r or r.
Let the object move from P to P′ through the path PAP′ in time t′- t.
Now the position vector of the object at time t′ is given by OP′ or r′ .
Displacement vector is PP′ or r′ - r .
Note that the displacement vector is same even if the object moves through
different courses from P to P′ , say PBP′ or any other infinite number of paths.
The position vector of the object provides following information:
(i) It gives straight line distance of the object from the origin.
(ii) It gives the direction of the object with respect to the origin.Home Next Previous
FEW FUNDAMENTAL DEFINITIONS IN VECTOR ALGEBRA
P
(i) Negative Vector A + (- A ) = 0
The negative of a vector is defined as another A O
vector having the same magnitude but drawn
in the opposite direction. -A
Example: The negative vector of A is - A
O
(ii) Equal Vectors
P
Two vectors A and B are said to be equal if, P
and only if, they have the same magnitude and S
the same direction. (Refer to previous slide) A B
(iii) Co-initial Vectors
Two vectors are said to be co-initial, if they
have a common initial point. O
O
A
(iv) Co-terminal Vectors
Two vectors are said to be co-terminal, if P
they terminate at a common point. B

(v) Co-planar Vectors S

The vectors are said to be co-planar, if they lie in the same
plane.
The vectors drawn in this slide are all co-planar vectors.
(vi) Collinear Vectors A″
Two vectors having equal or unequal O″ P″
magnitudes, which either act along the same B″
Q″ S″
line or along the parallel lines in the same
direction or along the parallel lines in A″
opposite direction are called collinear vectors. O″ P″
B″
Q″ S″
(vii) Unit Vector
A vector divided by its magnitude is called a unit
vector along the direction of the vector.
A
or A=
A unit vector is a vector of unit magnitude |A|
pointing in a particular direction.
Y
It has no dimension or unit. It is used to specify
a direction only.
Unit vectors along the x-, y- and z-axes of a
rectangular coordinate system are denoted by j
i, j, k respectively. X
k O
The magnitude of the unit vector is one i
unit and its direction is same as that of the
given vector.
Z
Since these are unit vectors,
|i|= |j|=|k|=1
 (viii) Zero Vector or Null Vector
Y A
Consider the position vectors in a plane
as shown in the figure. P′
P
Now suppose an object which is at P at
r B
time t, moves to P′ through the path r′
PAP′ and then comes back to P through
the path P′BP.
Then, what is its displacement? Find! O X

A VECTOR WHOSE INITIAL AND FINAL POSITIONS P

COINCIDE IS CALLED A ″NULL VECTOR″. O
A
The illustration of negative vector justifies
the need of a ′zero vector′. -A

A+0=0 +A=A O
λ0 =0 |0|=0 P
0A =0 A + (- A ) = 0

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MULTIPLICATION OF VECTORS BY REAL NUMBERS

Multiplying a vector A with a positive number λ gives a vector whose

magnitude is changed by the factor λ but the direction is the same as
that of A.
|λ A| = λ |A| if λ > 0

For example, if A is multiplied by 2, the resultant A 2A

vector 2A is in the same direction as A and has a
magnitude twice of |A| as shown in the figure.

Multiplying a vector A by a negative number λ

gives a vector λA whose direction is opposite to
the direction of A and whose magnitude is –λ
times |A|. A
|λ A| = -λ |A| if λ < 0
-A
Multiplying a given vector A by negative numbers,
say –1 and –1.5, gives vectors as shown in the -1.5A
figure.

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ADDITION AND SUBTRACTION OF VECTORS — GRAPHICAL METHOD
Addition of Vectors
Two or more vectors may be added graphically or geometrically by using
the following laws of vector addition:

1. Triangle Law of Vector Addition

If two vectors can be represented by the two sides of a triangle taken in the
same order, then the resultant is represented completely (both in magnitude
and direction) by the third side of the triangle taken in the opposite order.
Example:
Addition of two vectors A and B.
A P
To add vector B to vector A, move vector B
parallel to itself so as to place the tail of B O B
on head of A.
Join the tail of A to the head of B. Let this R
be vector OR or R. This gives the resultant
R or the sum of the vectors A and B.
This graphical method is called the ′head-to-tail method′.
The two vectors and their resultant form three sides of a triangle, so this
method is also known as triangle method of vector addition. Home Next Previous
2. Parallelogram Law of Vector Addition
If the two vectors (to be added) can be represented both in magnitude and
direction by the two adjacent sides of a parallelogram, then the resultant is
represented completely (both in magnitude and direction) by the diagonal of
the parallelogram passing through the starting point of the vectors.

Example:
Addition of two vectors A and B. P
A
To add vector B to vector A, move vector O
B parallel to itself so as to place the tail B
of B on tail of A.
R
Complete the parallelogram OQRP. Q
Draw the diagonal passing through the
common point O (meeting point of tails).

The diagonal represents the resultant or sum of the given two vectors.

Note that the resultant is the same as obtained by triangle law of vectors.

The two vectors are represented by the adjacent sides of a parallelogram and
the resultant is given by the diagonal, so this method is known as
parallelogram method of vector addition.
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Subtraction of Vectors
Subtraction of vectors can be defined by vector addition of one vector
with the negative vector of the other.
Example:
Subtraction of vector B from A

To subtract vector B from vector A, find

R
negative vector of B.
Displace –B parallel to itself so as to place
the tail of –B on head of A.
P
A
Join the tail of A to the head of -B. Let this O B-B
be vector OR or R. This gives the resultant
R or the difference of the vectors A and B.

A – B = A + (– B)

A + (– A ) = 0

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RESOLUTION OF VECTORS
The process of splitting up a vector into two or more components is known
as resolution of vector.

The vectors into which a given vector is split are called component vectors.
The component of a vector in a given direction gives the measure of the
effect of the vector in that direction.
Rectangular Resolution of a Vector in a Plane
We can resolve a vector in terms of component vectors that lie along unit
vectors i and j .
Consider a vector A that lies in x-y plane Y
as shown in the figure.
A2
Draw lines from the head of A perpendicular to the
coordinate axes. Ay j
A
We get, vectors A1 and A2 such that A1 + A2 = A. X
O Ax i A1
Since A1 is parallel to i and A2 is parallel to j ,
we have A1 = Axi and A2 = Ay j where Ax and Ay are real numbers.
The quantities Ax and Ay are called x-, and y- components of the vector A.
A = Axi + Ay j Note Ax and Ay are scalars. Home Next Previous
Resolution of a Vector Using Trigonometry
Using simple trigonometry, we can express Ax and Ay in terms of the
magnitude of A and the angle θ it makes with the x-axis:
Y
Ax = A cos θ
Ay = A sin θ
A2
A component of a vector can be positive,
Ay j
negative or zero depending on the value of θ. A θ
Now, we have two ways to specify a vector A in X
O Ax i A1
a plane. It can be specified by:
(i) its magnitude A and the direction θ it makes with the x-axis; or
(ii) its components Ax and Ay
If ′A′ and ′θ′ are given, then Ax and Ay can be obtained using the above
equation.

If Ax and Ay are given, ′A′ and ′θ′ can be obtained as follows:

Ax2 + Ay2 = A2 cos2 θ + A2 sin2 θ = A2
or A = Ax2 + Ay2
Ay Ay
and tan θ = or θ= tan-1
Ax Ax Home Next Previous
Resolution of a Vector into 3 Rectangular components in Space (3-D)
Let us resolve a general vector A into three components along x-, y-, and z-
axes in three dimensions. Y
If α, β, and γ are the angles between A and the x-,
y-, and z-axes respectively, then
Ax
Ax = A cos α, cos α = Ay j
A
Ay = A cos β, | A | γ α
Az = A cos γ Ay X
cos β = O Ax i
Az k
|A|
Az Z
cos γ =
|A| α, β, and γ are the angles in
space. They are the angles
In general, we have A = Axi + Ay j + Az k between pairs of lines which
are not coplanar.
The magnitude of vector A is A = Ax2 + Ay2 + Az2
A position vector r can be expressed as

r = xi + y j + z k Also, cos2 α + cos2 β + cos2 γ = 1

where x, y, and z are the components of r along

x-, y-, z-axes, respectively. Home Next Previous
VECTOR ADDITION IN 3-D – ANALYTICAL METHOD
Consider two vectors A and B in x-y-z coordinate system representing space
(3-D) with components Ax, Ay, Az and Bx, By, Bz :

A = Axi + Ay j + Az k
B = Bxi + Byj + Bzk

Let R be the sum. Then R = A + B

R = Axi + Ay j + Az k + Bxi + Byj + Bzk

Since vectors obey the commutative and associative

laws,
R = (Ax + Bx) i + (Ay + By) j + (Az + Bz) k
Since R = Rxi + Ry j + Rz k

Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz

Thus, each component of the resultant vector R is the sum of the

corresponding components of A and B.

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This method can be extended to addition and subtraction of any number of
vectors. For example, if vectors a, b and c are given as

a = axi + ay j + az k

b = bxi + by j + bz k

c = cx i + cy j + cz k

then, a vector T = a + b – c has components:

Tx = ax + bx - cx

Ty = ay + by - cy

Tz = az + bz - cz

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]

PRODUCT OF TWO VECTORS

Two types of product of vectors are:
1. Dot or Scalar Product and
2. Cross or Vector Product

1. Dot Product or Scalar Product

In dot product, the symbol of multiplication between the vectors is
represented by ‘ . ‘ The result of the product is a scalar value.

Eg.
Work is defined as the product of force and displacement.
Here force and displacement are vectors and work is a scalar.
P
Consider two vectors A and B making an angle θ
with each other. A
The dot or scalar product is given by
θ
A . B = | A | | B | cos θ O
B Q
A.B
and cos θ =
|A||B| Home Next Previous
Properties of Dot Product
1. Dot product results in a scalar value.
2. Dot product or Scalar product is commutative. A.B=B.A

3. Dot product is distributive. A . (B + C) = A . B + A . C

4. i . i = j . j = k . k = 1

5. i . j = j . k = k . i = 0
A.B
6. Component of A along B = | A | cos θ = =A.B
|B|

A.B
7. Component of B along A = | B | cos θ = =B.A
|A|
8. If a = axi + ay j + az k

b = bxi + by j + bz k

then a . b = (ax bx + ay by + az bz)

a . a = (ax ax + ay ay + az az) a2 = (ax2 + ay2 + az2)
b . b = (bx bx + by by + bz bz) b2 = (bx2 + by2 + bz2)
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2. Cross Product or Vector Product
In vector product, the symbol of multiplication between the vectors
is represented by ‘ x ‘. The result of the product is a vector.

Eg. Torque is defined as the product of force and moment arm.

Here force, moment arm and torque are all vectors. P
Consider two vectors A and B making an angle θ
with each other. A
The cross or vector product is given by
θ
A x B = | A | | B | sinnθ where n is the unit vector O
along a direction which is B Q
|A x B| perpendicular to plane
and sin θ = containing A and B.
|A||B|
A x B means the vector A is rotated towards B and its effect is taken on B.
By Right Hand Thumb Rule or Maxwell’s Cork Screw Rule, the direction
of the resultant in this case is perpendicular and into the plane of the
diagram.
B x A means the vector B is rotated towards A and its effect is taken on A.
The resultant in this case is perpendicular and emerging out of the plane of
the diagram.
Closing and opening a tap is the best example. Home Next Previous
Properties of Cross
Product
1. Vector product results in a vector.
2. Vector product or Cross product is not commutative. AxB=-BxA

3. ix i = jxj = kxk = 0

4. i x j = k 5. j x i = - k
jx k = i kx j = - i
k xi = j i xk=-j

6. If a = axi + ay j + az k

b = bxi + by j + bz k

axb= i j k = (ay bz – az by) i – (ax bz – az bx) j + (ax by – aybx) k

ax ay az

bx by bz

| a x b | = (ay bz – az by)2 + (ax bz – az bx)2 + (ax by – aybx)2

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VELOCITY A particle moves along the curved path as shown. At time t 1
its position is ri and at time t2 its position is rf.
y

The instantaneous
vi velocity points
r vf tangent to the path.

ri rf

r
v av  Points in the direction of r
t
 A displacement over an interval of
time is a velocity
r  x 
Average velocity  v av   The x - component would be : vav , x  
t  t 

r
Instantane ous velocity  v  lim
t 0 t

The instantaneous velocity is represented by the slope

of a line tangent to the curve on the graph of an
object’s position versus time.
 A particle moves along the curved path as shown. At time t 1
A its position is r0 and at time t2 its position is rf.
C y
C a av 
v Points in the
E t direction of v.
vi
L v
E vf

R ri rf
A
T
I x
O
N
A nonzero acceleration changes an
object’s state of motion

v
Average accelerati on  a av 
t

Δv
Instantaneous acceleration = a = lim
Δt 0 Δt

These have interpretations similar to vav and v.

PROJECTILE MOTION
EQUATION OF TRAJECTORY OF INCLINED
PROJECTILE
TIME OF FLIGHT, HORIZONTAL RANGE AND MAXIMUM
HEIGHT OF AN INCLINED PROJECTILE
UNIFORM CIRCULAR MOTION
UNIFORM CIRCULAR MOTION
4. A cricket ball is thrown at a speed of 28m/s in a direction 30
above the horizontal. Calculate (i) Maximum height (ii) The time
AFL
taken by the ball to return to the same level (iii) The distance
from the thrower to the point where the ball returns to the same
level.
5
6 An arrow is shot into the air with  = 60° and vi = 20.0 m/s.
(a) What are vx and vy of the arrow when t = 3 sec?

y
The components of the initial
vi velocity are:

60° vix  vi cos   10.0 m/s

x
viy  vi sin   17.3 m/s

v fx  vix  ax t  vix  10.0 m/s CONSTANT

At t = 3 sec:
v fy  viy  a y t  viy  gt  12.1 m/s
(b) What are the x and y components of the displacement
of the arrow during the 3.0 sec interval?
y

x
1
Δrx = Δx = x f - xi = vix Δt + a x Δt 2 = vix Δt + 0 = 30.0 m
2
1 1
Δry = Δy = y f - yi = viy Δt + a y Δt 2 = viy Δt - gΔt 2 = 7.80 m
2 2
7 How far does the arrow in the previous example land from where it is
released?

1
The arrow lands when y = 0. y  viy t  gt 2  0
2
1
Δy = (viy - gΔt)t = 0
2

2viy
Solving for t: t   3.53 sec
g

The distance traveled is: Δx = vix Δt = 35.3 m