Motion in a
4 Plane
SCALARS AND VECTORS (v) Unit vector : Vector which has unit magnitude. It represents
Scalars : The physical quantities which have only magnitude r
direction only. For example take a vector B . Unit vector in
but no direction, are called scalar quantities.
For example - distance, speed, work, temperature, mass, etc. r B
· Scalars are added, subtracted, multiplied and divided by the direction of B is , which is denoted as B̂ . B̂ , is
|B |
ordinary laws of algebra. read as “B cap” or "B caret".
Vectors: For any quantity to be a vector, (vi) Orthogonal unit vector : A set of unit vectors, having the
(i) it must have magnitude.
directions of the positive x, y and z axes of three dimensional
(ii) it must have direction.
(iii) it must satisfy parallelogram law of vector addition. rectangular coordinate system are denoted by ˆi, ˆj and kˆ .
For example – displacement, velocity, force, etc. They are called orthogonal unit vectors because angle
Electric current has magnitude as well as direction but between any of the two unit vectors is 90º.
still it is not treated as a vector quantity because it is added by Z
ordinary law of algebra.
k̂
Types of Vectors
(i) Like vectors : Vectors having same direction are called like
vectors. The magnitude may or may not be equal. ˆj
o Y
A î
B
r r
A and B are like vectors. These are also called parallel X
vectors or collinear vectors.
The coordinate system which has shown in fig. is called
(ii) Equal vectors : Vectors having same magnitude and same
direction are called equal vectors. right handed coordinate system. Such a system derives its
name from the fact that right threaded screw rotated through
A 90º from OX to OY will advance in positive Z direction as
B shown in the figure.
r r r r
Here A and B are equal vectors A = B (vii) Null vector (zero vector) : A vector of zero magnitude is
Thus, equal vector is a special case of like vector. called a zero or null vector. Its direction is not defined. It is
(iii) Unlike vectors : Vectors having exactly opposite directions denoted by 0.
are called unlike vectors. The magnitude may or may not be Properties of Null or Zero Vector :
equal. ur
(a) The sum of a finite vector A and the zero vector is
A
B equal to the finite vector
r r ur ur
A and B are unlike vectors. i.e., A + 0 = A
(iv) Negative vectors : Vectors having exactly opposite direction (b) The multiplication of a zero vector by a finite number n
and equal magnitudes are called negative vectors. is equal to the zero vector
i.e., 0 n = 0
A ur
(c) The multiplication of a finite A by a zero is equal to
B zero vector
r r r r ur
Here A and B are negative vectors, A = -B i.e., A 0 = 0
Thus negative vectors is a special case of unlike vectors.
, 68 PHYSICS
(viii) Axial vector : Vector associated with rotation about an axis ur ur
Direction of R : Let the resultant R makes an angle b with the
i.e., produce rotation effect is called axial vector. Examples ur
are angular velocity, angular momentum, torque etc. direction of A . Then from right angle triangle QNO,
(ix) Coplanar vectors : Vectors in the same plane are called QN QN B sin q
coplanar vectors. tan b = = =
ON OP + PN A + B cos q
(x) Position vectors and displacement vectors : The vector ur
drawn from the origin of the co-ordinate axes to the position (i) | R | is maximum, if cosq = 1, q = 0° (parallel vector)
ur ur
of a particle is called position vector of the particle. If A (x1, A B
y1, z1) and B (x2, y2, z2) be the positions of the particle at ¾¾® P ¾¾®
two different times of its motion w.r.t. the origin O, then
Rmax = A 2 + B2 + 2AB = A + B
position vector of A and B are ur
Y (ii) | R | is minimum, if cosq = –1, q = 180° (opposite vector)
uur uuur
B¾ ¾¾
¬¾ A®
Pa
A D
th
Rmin = A 2 + B2 - 2AB = A - B
of
isp
ve lace
pa
rA (iii) If the vectors A and B are orthogonal,
rti
c mto
r ent
cl
e
B i.e., q = 90 o , R = A 2 + B2
rB Parallelogram Law of Vector Addition
O X
It states that if two vectors are represented in magnitude and
uur uuur
rA = OA = x1 ˆi + y1 ˆj + z1 kˆ direction by the two adjacent sides of a parallelogram then their
uur uuur resultant is represented in magnitude and direction by the
rB = OB = x 2 ˆi + y2 ˆj + z 2 kˆ . diagonal of the parallelogram.
ur ur
The displacement vector is Let the two vectors A and B , inclined at angle q are represented
uuur uuur
AB = OB - OA . by sides OP and OS of parallelogram OPQS, then resultant vector
ur uuur
= (x 2 - x 2 )iˆ + (y 2 - y1 )ˆj + (z 2 - z1 )kˆ R is represented by diagonal OQ of the parallelogram.
Laws of Vector Algebra S Q
r r r r
1. A + B = B + A (Commutative law of addition)
r r r r r r
2. A + (B + C) = (A + B) + C (Associative law of addition) R
r r B
3. mA = Am q
r r b
4. m(nA) = (mn)A O
P
r r r A
5. (m + n)A = mA + nA B sin q
r r r r R = A2 + B 2 + 2 AB cos q ; tan b =
6. m(A + B) = mA + mB A + B cos q
ur ur ur ur
ADDITION OF VECTORS If q < 90° , (acute angle) R = A + B , R is called main
(major) diagonal of parallelogram
Triangle Law of Vector Addition ur ur ur ur
It states that if two vectors acting on a particle at the same time If q > 90° , (obtuse angle) R = A + B , R is called minor
diagonal.
are represented in magnitude and direction by the two sides of a
triangle taken in one order, their resultant vector is represented Polygon Law of Vector Addition
in magnitude and direction by the third side of the triangle taken If a number of non zero vectors are represented by the (n–1)
in opposite order. sides of an n-sided polygon then the resultant is given by the
closing side or the nth side of the polygon taken in opposite
Q
order. D D C
C
E
R B B
E
q
b R B
O A P N O A
A
ur ur ur ur ur ur
ur So, R = A + B + C + D + E
Magnitude of R is given by R = A 2 + B 2 + 2 AB cos q uuur uuur uuur uuur uuur uuur
ur ur or, OA + AB + BC + CD + DE = OE
where q is the angle between A and B .
, Motion in a Plane 69
15. Magnitude of a vector is independent of co-ordinate axes
system.
1. Resultant of two unequal vectors cannot be zero. 16. Component of a vector perpendicular to itself is zero.
2. Resultant of three co-planar vectors may or may not be 17. (a) Resultant of two vectors is maximum when angle
zero. between the vectors is zero i.e., q = 0°
3. Minimum no. of coplanar vectors for zero resultant is 2 (for Rmax = A + B
(b) Resultant of two vectors is minimum when
equal magnitude) and 3 (for unequal magnitude).
q = 180°
4. Resultant of three non coplanar vectors cannot be zero. Rmin = A – B
Minimum number of non coplanar vectors whose sum can
be zero is four. (c) The magnitude of resultant of A and B can vary
between (A + B) and (A – B)
5. Polygon law should be used only for diagram purpose for
calculation of resultant vector (For addition of more than 2 SUBTRACTION OF VECTORS
vectors) we use components of vector. We convert vector subtraction into vector addition.
Keep in Memory B
r r r r q
1. If A = B , then A - B = 0 is a null vector..
2. Null vector or zero vector is defined as a vector whose A
–B 180° – q
magnitude is zero and direction indeterminate. Null vector
differs from ordinary zero in the sense that ordinary zero is
not associated with direction. A - B = A + (-B )
r
A If the angle between A and B is q then the angle between
3. Â = r is called a unit vector. It is unitless and
|A|
A
dimensionless vector. Its magnitude is 1. It represents
direction only. a
r r r r
180
4. If A = B , then | A |=| B | and  = B̂ , where Aˆ and B ˆ are –B
°–q
unit vectors of A and B respectively.
5. A vector can be divided or multiplied by a scalar. R
6. Vectors of the same kind can only be added or subtracted.
A and - B is (180° – q).
It is not possible to add or subtract the vectors of different
kind. This rule is also valid for scalars.
| A - B |= A 2 + B2 - 2AB cos q
7. Vectors of same as well as different kinds can be multiplied.
8. A vector can have any number of components. But it can B sin(180 o - q) B sin q
have only three rectangular components in space and two tan a = =
A + B cos (180o - q) A - B cos q
rectangular components in a plane. Rectangular
components are mutually perpendicular. RESOLUTION OF A VECTOR
9. The minimum number of unequal non-coplanar whose
vector sum is zero is 4. Rectangular Components of a Vector in Plane
ur Y
10. When A = A x ˆi + A y ˆj + A z kˆ
r r
| A |= A 2x + A 2y + A z2 , where | A | is modulus or
A y ˆj A y ˆj
r A
magnitude of vector A .
11. ˆi + ˆj makes 45° with both X and Y-axes. It makes angle 90°
q
with Z-axis.
A x î X
12. î + ˆj + k̂ makes angle 54.74° with each of the X, Y and
Z-axes. The vector A may be written as
uur
13. A -B ¹ B -A A = Axiˆ + Ay ˆj
14. If | A + B | = | A - B | then angle between A and B is
where A x î is the component of vector A in X-direction and
p
.
2 A y ĵ is the component of vector A in the Y-direction.
, EBD_7751
70 PHYSICS
Also Ax = A cos q and Ay = A sin q Solution :
\ A = ( A cos q) î + ( A sin q) ĵ Here OP = 6 cos 60° î + 6 sin 60 ° ĵ = 3 î + 5.2 ĵ `
Þ A cos q and A sin q are the magnitudes of the components of PQ = 4 î and QM = 3 ˆj
A in X and Y-direction respectively.. \ OM = OP + PQ + QM = 7 î + 8.2 ˆj
uur Example 3.
Also | A |= Ax2 + Ay2 The resultant of two forces F1 and F2 is P. If F2 is reversed,
Rectangular components of a vector in 3D : Three rectangular the resultant is Q. Show that P 2 + Q 2 = 2( F12 + F22 ) .
components along X, Y and Z direction are given by
A x ˆi , A y ˆj, A z k.
ˆ Therefore, F1
r
vector A may be written as
ur F2 P F2
q
Q
A = Ax ˆi + A y ˆj + A z kˆ and A = A 2x + A 2y + A 2z
q
If a , b and g are the angles subtended by the rectangular a
components of vector then F1
A Ay Az Solution :
cos a = x , cos b = and cos g = r r
A A A Suppose q be the angle between the forces F1 and F2 ,
2 2
Also, cos a + cos b + cos 2 g =1 then P 2 = F12 + F22 + 2F1F2 cos q ......(i)
CAUTION : Do not resolve the vector at its head. The vector
is always resolved at its tail. F1
B cos q
q B sin q Q
P
F2 F2
B
q
B
B sin q q
F1
B cos q
Wrong Correct When the force F2 is reversed, then the magnitude of their
Example 1. resultant is
X and Y component of vector A are 4 and 6 m respectively. Q 2 = F12 + F22 + 2 F1 F2 cos(180° - q)
uur uur
The X and Y component of A + B are 10 m and 9 m = F12 + F22 - 2F1F2 cos q ......(ii)
respectively. Calculate the length of vector B and its angle
with respect to X-axis Adding equations (i) and (ii),
Solution : P 2 + Q 2 = 2F12 + 2F22 = 2(F12 + F22 )
A = 4 î + 6 ˆj and A + B = 10 î + 9 ˆj Example 4.
r
Find the components of vector A = 2iˆ + 3jˆ along the
\ B = ( A + B ) – A = (10î + 9 ĵ) – ( 4 î + 6 ˆj) = 6 î + 3 ĵ
directions of ˆi + ˆj and ˆi – ˆj .
\ length of B is | B |= 6 2 + 3 2 = 3 5 m Solution :
r
By
3 1 æ1ö Here A = 2î + 3ˆj
Also tan q = = Þ q = tan -1 ç ÷
= r
Bx
6 2 è2ø In order to find the component of A along the direction of
r r
where q is the angle which (A + B) is making with î + ĵ , let us find out the unit vector along î + ĵ . If â is the
X-axis.
Example 2. unit vector along î + ĵ , then
Find the resultant of vectors given in figure
î + ˆj î + ˆj
Y M â = =
| î + ˆj | 2
3 cm r
Hence, the magnitude of the component vector of A along
P 4 cm
Q î + ĵ
6cm
r î + ĵ 1 5
60° = A.â = (2î + 3ˆj). = (2 + 3) =
X 2 2 2
O
4 Plane
SCALARS AND VECTORS (v) Unit vector : Vector which has unit magnitude. It represents
Scalars : The physical quantities which have only magnitude r
direction only. For example take a vector B . Unit vector in
but no direction, are called scalar quantities.
For example - distance, speed, work, temperature, mass, etc. r B
· Scalars are added, subtracted, multiplied and divided by the direction of B is , which is denoted as B̂ . B̂ , is
|B |
ordinary laws of algebra. read as “B cap” or "B caret".
Vectors: For any quantity to be a vector, (vi) Orthogonal unit vector : A set of unit vectors, having the
(i) it must have magnitude.
directions of the positive x, y and z axes of three dimensional
(ii) it must have direction.
(iii) it must satisfy parallelogram law of vector addition. rectangular coordinate system are denoted by ˆi, ˆj and kˆ .
For example – displacement, velocity, force, etc. They are called orthogonal unit vectors because angle
Electric current has magnitude as well as direction but between any of the two unit vectors is 90º.
still it is not treated as a vector quantity because it is added by Z
ordinary law of algebra.
k̂
Types of Vectors
(i) Like vectors : Vectors having same direction are called like
vectors. The magnitude may or may not be equal. ˆj
o Y
A î
B
r r
A and B are like vectors. These are also called parallel X
vectors or collinear vectors.
The coordinate system which has shown in fig. is called
(ii) Equal vectors : Vectors having same magnitude and same
direction are called equal vectors. right handed coordinate system. Such a system derives its
name from the fact that right threaded screw rotated through
A 90º from OX to OY will advance in positive Z direction as
B shown in the figure.
r r r r
Here A and B are equal vectors A = B (vii) Null vector (zero vector) : A vector of zero magnitude is
Thus, equal vector is a special case of like vector. called a zero or null vector. Its direction is not defined. It is
(iii) Unlike vectors : Vectors having exactly opposite directions denoted by 0.
are called unlike vectors. The magnitude may or may not be Properties of Null or Zero Vector :
equal. ur
(a) The sum of a finite vector A and the zero vector is
A
B equal to the finite vector
r r ur ur
A and B are unlike vectors. i.e., A + 0 = A
(iv) Negative vectors : Vectors having exactly opposite direction (b) The multiplication of a zero vector by a finite number n
and equal magnitudes are called negative vectors. is equal to the zero vector
i.e., 0 n = 0
A ur
(c) The multiplication of a finite A by a zero is equal to
B zero vector
r r r r ur
Here A and B are negative vectors, A = -B i.e., A 0 = 0
Thus negative vectors is a special case of unlike vectors.
, 68 PHYSICS
(viii) Axial vector : Vector associated with rotation about an axis ur ur
Direction of R : Let the resultant R makes an angle b with the
i.e., produce rotation effect is called axial vector. Examples ur
are angular velocity, angular momentum, torque etc. direction of A . Then from right angle triangle QNO,
(ix) Coplanar vectors : Vectors in the same plane are called QN QN B sin q
coplanar vectors. tan b = = =
ON OP + PN A + B cos q
(x) Position vectors and displacement vectors : The vector ur
drawn from the origin of the co-ordinate axes to the position (i) | R | is maximum, if cosq = 1, q = 0° (parallel vector)
ur ur
of a particle is called position vector of the particle. If A (x1, A B
y1, z1) and B (x2, y2, z2) be the positions of the particle at ¾¾® P ¾¾®
two different times of its motion w.r.t. the origin O, then
Rmax = A 2 + B2 + 2AB = A + B
position vector of A and B are ur
Y (ii) | R | is minimum, if cosq = –1, q = 180° (opposite vector)
uur uuur
B¾ ¾¾
¬¾ A®
Pa
A D
th
Rmin = A 2 + B2 - 2AB = A - B
of
isp
ve lace
pa
rA (iii) If the vectors A and B are orthogonal,
rti
c mto
r ent
cl
e
B i.e., q = 90 o , R = A 2 + B2
rB Parallelogram Law of Vector Addition
O X
It states that if two vectors are represented in magnitude and
uur uuur
rA = OA = x1 ˆi + y1 ˆj + z1 kˆ direction by the two adjacent sides of a parallelogram then their
uur uuur resultant is represented in magnitude and direction by the
rB = OB = x 2 ˆi + y2 ˆj + z 2 kˆ . diagonal of the parallelogram.
ur ur
The displacement vector is Let the two vectors A and B , inclined at angle q are represented
uuur uuur
AB = OB - OA . by sides OP and OS of parallelogram OPQS, then resultant vector
ur uuur
= (x 2 - x 2 )iˆ + (y 2 - y1 )ˆj + (z 2 - z1 )kˆ R is represented by diagonal OQ of the parallelogram.
Laws of Vector Algebra S Q
r r r r
1. A + B = B + A (Commutative law of addition)
r r r r r r
2. A + (B + C) = (A + B) + C (Associative law of addition) R
r r B
3. mA = Am q
r r b
4. m(nA) = (mn)A O
P
r r r A
5. (m + n)A = mA + nA B sin q
r r r r R = A2 + B 2 + 2 AB cos q ; tan b =
6. m(A + B) = mA + mB A + B cos q
ur ur ur ur
ADDITION OF VECTORS If q < 90° , (acute angle) R = A + B , R is called main
(major) diagonal of parallelogram
Triangle Law of Vector Addition ur ur ur ur
It states that if two vectors acting on a particle at the same time If q > 90° , (obtuse angle) R = A + B , R is called minor
diagonal.
are represented in magnitude and direction by the two sides of a
triangle taken in one order, their resultant vector is represented Polygon Law of Vector Addition
in magnitude and direction by the third side of the triangle taken If a number of non zero vectors are represented by the (n–1)
in opposite order. sides of an n-sided polygon then the resultant is given by the
closing side or the nth side of the polygon taken in opposite
Q
order. D D C
C
E
R B B
E
q
b R B
O A P N O A
A
ur ur ur ur ur ur
ur So, R = A + B + C + D + E
Magnitude of R is given by R = A 2 + B 2 + 2 AB cos q uuur uuur uuur uuur uuur uuur
ur ur or, OA + AB + BC + CD + DE = OE
where q is the angle between A and B .
, Motion in a Plane 69
15. Magnitude of a vector is independent of co-ordinate axes
system.
1. Resultant of two unequal vectors cannot be zero. 16. Component of a vector perpendicular to itself is zero.
2. Resultant of three co-planar vectors may or may not be 17. (a) Resultant of two vectors is maximum when angle
zero. between the vectors is zero i.e., q = 0°
3. Minimum no. of coplanar vectors for zero resultant is 2 (for Rmax = A + B
(b) Resultant of two vectors is minimum when
equal magnitude) and 3 (for unequal magnitude).
q = 180°
4. Resultant of three non coplanar vectors cannot be zero. Rmin = A – B
Minimum number of non coplanar vectors whose sum can
be zero is four. (c) The magnitude of resultant of A and B can vary
between (A + B) and (A – B)
5. Polygon law should be used only for diagram purpose for
calculation of resultant vector (For addition of more than 2 SUBTRACTION OF VECTORS
vectors) we use components of vector. We convert vector subtraction into vector addition.
Keep in Memory B
r r r r q
1. If A = B , then A - B = 0 is a null vector..
2. Null vector or zero vector is defined as a vector whose A
–B 180° – q
magnitude is zero and direction indeterminate. Null vector
differs from ordinary zero in the sense that ordinary zero is
not associated with direction. A - B = A + (-B )
r
A If the angle between A and B is q then the angle between
3. Â = r is called a unit vector. It is unitless and
|A|
A
dimensionless vector. Its magnitude is 1. It represents
direction only. a
r r r r
180
4. If A = B , then | A |=| B | and  = B̂ , where Aˆ and B ˆ are –B
°–q
unit vectors of A and B respectively.
5. A vector can be divided or multiplied by a scalar. R
6. Vectors of the same kind can only be added or subtracted.
A and - B is (180° – q).
It is not possible to add or subtract the vectors of different
kind. This rule is also valid for scalars.
| A - B |= A 2 + B2 - 2AB cos q
7. Vectors of same as well as different kinds can be multiplied.
8. A vector can have any number of components. But it can B sin(180 o - q) B sin q
have only three rectangular components in space and two tan a = =
A + B cos (180o - q) A - B cos q
rectangular components in a plane. Rectangular
components are mutually perpendicular. RESOLUTION OF A VECTOR
9. The minimum number of unequal non-coplanar whose
vector sum is zero is 4. Rectangular Components of a Vector in Plane
ur Y
10. When A = A x ˆi + A y ˆj + A z kˆ
r r
| A |= A 2x + A 2y + A z2 , where | A | is modulus or
A y ˆj A y ˆj
r A
magnitude of vector A .
11. ˆi + ˆj makes 45° with both X and Y-axes. It makes angle 90°
q
with Z-axis.
A x î X
12. î + ˆj + k̂ makes angle 54.74° with each of the X, Y and
Z-axes. The vector A may be written as
uur
13. A -B ¹ B -A A = Axiˆ + Ay ˆj
14. If | A + B | = | A - B | then angle between A and B is
where A x î is the component of vector A in X-direction and
p
.
2 A y ĵ is the component of vector A in the Y-direction.
, EBD_7751
70 PHYSICS
Also Ax = A cos q and Ay = A sin q Solution :
\ A = ( A cos q) î + ( A sin q) ĵ Here OP = 6 cos 60° î + 6 sin 60 ° ĵ = 3 î + 5.2 ĵ `
Þ A cos q and A sin q are the magnitudes of the components of PQ = 4 î and QM = 3 ˆj
A in X and Y-direction respectively.. \ OM = OP + PQ + QM = 7 î + 8.2 ˆj
uur Example 3.
Also | A |= Ax2 + Ay2 The resultant of two forces F1 and F2 is P. If F2 is reversed,
Rectangular components of a vector in 3D : Three rectangular the resultant is Q. Show that P 2 + Q 2 = 2( F12 + F22 ) .
components along X, Y and Z direction are given by
A x ˆi , A y ˆj, A z k.
ˆ Therefore, F1
r
vector A may be written as
ur F2 P F2
q
Q
A = Ax ˆi + A y ˆj + A z kˆ and A = A 2x + A 2y + A 2z
q
If a , b and g are the angles subtended by the rectangular a
components of vector then F1
A Ay Az Solution :
cos a = x , cos b = and cos g = r r
A A A Suppose q be the angle between the forces F1 and F2 ,
2 2
Also, cos a + cos b + cos 2 g =1 then P 2 = F12 + F22 + 2F1F2 cos q ......(i)
CAUTION : Do not resolve the vector at its head. The vector
is always resolved at its tail. F1
B cos q
q B sin q Q
P
F2 F2
B
q
B
B sin q q
F1
B cos q
Wrong Correct When the force F2 is reversed, then the magnitude of their
Example 1. resultant is
X and Y component of vector A are 4 and 6 m respectively. Q 2 = F12 + F22 + 2 F1 F2 cos(180° - q)
uur uur
The X and Y component of A + B are 10 m and 9 m = F12 + F22 - 2F1F2 cos q ......(ii)
respectively. Calculate the length of vector B and its angle
with respect to X-axis Adding equations (i) and (ii),
Solution : P 2 + Q 2 = 2F12 + 2F22 = 2(F12 + F22 )
A = 4 î + 6 ˆj and A + B = 10 î + 9 ˆj Example 4.
r
Find the components of vector A = 2iˆ + 3jˆ along the
\ B = ( A + B ) – A = (10î + 9 ĵ) – ( 4 î + 6 ˆj) = 6 î + 3 ĵ
directions of ˆi + ˆj and ˆi – ˆj .
\ length of B is | B |= 6 2 + 3 2 = 3 5 m Solution :
r
By
3 1 æ1ö Here A = 2î + 3ˆj
Also tan q = = Þ q = tan -1 ç ÷
= r
Bx
6 2 è2ø In order to find the component of A along the direction of
r r
where q is the angle which (A + B) is making with î + ĵ , let us find out the unit vector along î + ĵ . If â is the
X-axis.
Example 2. unit vector along î + ĵ , then
Find the resultant of vectors given in figure
î + ˆj î + ˆj
Y M â = =
| î + ˆj | 2
3 cm r
Hence, the magnitude of the component vector of A along
P 4 cm
Q î + ĵ
6cm
r î + ĵ 1 5
60° = A.â = (2î + 3ˆj). = (2 + 3) =
X 2 2 2
O
