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1 | 5018-5021 | However, an answer to the second query is a quantity (called
force) which involves muscular strength (magnitude) and
direction (in which another player is positioned) Such
quantities are called vectors In mathematics, physics and
engineering, we frequently come across with both types of
quantities, namely, scalar quantities such as length, mass,
time, distance, speed, area, volume, temperature, work,
money, voltage, density, resistance etc and vector quantities like displacement, velocity,
acceleration, force, weight, momentum, electric field intensity etc |
1 | 5019-5022 | Such
quantities are called vectors In mathematics, physics and
engineering, we frequently come across with both types of
quantities, namely, scalar quantities such as length, mass,
time, distance, speed, area, volume, temperature, work,
money, voltage, density, resistance etc and vector quantities like displacement, velocity,
acceleration, force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various
operations on vectors, and their algebraic and geometric properties |
1 | 5020-5023 | In mathematics, physics and
engineering, we frequently come across with both types of
quantities, namely, scalar quantities such as length, mass,
time, distance, speed, area, volume, temperature, work,
money, voltage, density, resistance etc and vector quantities like displacement, velocity,
acceleration, force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various
operations on vectors, and their algebraic and geometric properties These two type of
properties, when considered together give a full realisation to the concept of vectors,
and lead to their vital applicability in various areas as mentioned above |
1 | 5021-5024 | and vector quantities like displacement, velocity,
acceleration, force, weight, momentum, electric field intensity etc In this chapter, we will study some of the basic concepts about vectors, various
operations on vectors, and their algebraic and geometric properties These two type of
properties, when considered together give a full realisation to the concept of vectors,
and lead to their vital applicability in various areas as mentioned above 10 |
1 | 5022-5025 | In this chapter, we will study some of the basic concepts about vectors, various
operations on vectors, and their algebraic and geometric properties These two type of
properties, when considered together give a full realisation to the concept of vectors,
and lead to their vital applicability in various areas as mentioned above 10 2 Some Basic Concepts
Let ‘l’ be any straight line in plane or three dimensional space |
1 | 5023-5026 | These two type of
properties, when considered together give a full realisation to the concept of vectors,
and lead to their vital applicability in various areas as mentioned above 10 2 Some Basic Concepts
Let ‘l’ be any straight line in plane or three dimensional space This line can be given
two directions by means of arrowheads |
1 | 5024-5027 | 10 2 Some Basic Concepts
Let ‘l’ be any straight line in plane or three dimensional space This line can be given
two directions by means of arrowheads A line with one of these directions prescribed
is called a directed line (Fig 10 |
1 | 5025-5028 | 2 Some Basic Concepts
Let ‘l’ be any straight line in plane or three dimensional space This line can be given
two directions by means of arrowheads A line with one of these directions prescribed
is called a directed line (Fig 10 1 (i), (ii)) |
1 | 5026-5029 | This line can be given
two directions by means of arrowheads A line with one of these directions prescribed
is called a directed line (Fig 10 1 (i), (ii)) Chapter 10
VECTOR ALGEBRA
W |
1 | 5027-5030 | A line with one of these directions prescribed
is called a directed line (Fig 10 1 (i), (ii)) Chapter 10
VECTOR ALGEBRA
W R |
1 | 5028-5031 | 1 (i), (ii)) Chapter 10
VECTOR ALGEBRA
W R Hamilton
(1805-1865)
© NCERT
not to be republished
VECTOR ALGEBRA
425
Now observe that if we restrict the line l to the line segment AB, then a magnitude
is prescribed on the line l with one of the two directions, so that we obtain a directed
line segment (Fig 10 |
1 | 5029-5032 | Chapter 10
VECTOR ALGEBRA
W R Hamilton
(1805-1865)
© NCERT
not to be republished
VECTOR ALGEBRA
425
Now observe that if we restrict the line l to the line segment AB, then a magnitude
is prescribed on the line l with one of the two directions, so that we obtain a directed
line segment (Fig 10 1(iii)) |
1 | 5030-5033 | R Hamilton
(1805-1865)
© NCERT
not to be republished
VECTOR ALGEBRA
425
Now observe that if we restrict the line l to the line segment AB, then a magnitude
is prescribed on the line l with one of the two directions, so that we obtain a directed
line segment (Fig 10 1(iii)) Thus, a directed line segment has magnitude as well as
direction |
1 | 5031-5034 | Hamilton
(1805-1865)
© NCERT
not to be republished
VECTOR ALGEBRA
425
Now observe that if we restrict the line l to the line segment AB, then a magnitude
is prescribed on the line l with one of the two directions, so that we obtain a directed
line segment (Fig 10 1(iii)) Thus, a directed line segment has magnitude as well as
direction Definition 1 A quantity that has magnitude as well as direction is called a vector |
1 | 5032-5035 | 1(iii)) Thus, a directed line segment has magnitude as well as
direction Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10 |
1 | 5033-5036 | Thus, a directed line segment has magnitude as well as
direction Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB
uuur
or
simply as ar , and read as ‘vector AB
uuur
’ or ‘vector ar ’ |
1 | 5034-5037 | Definition 1 A quantity that has magnitude as well as direction is called a vector Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB
uuur
or
simply as ar , and read as ‘vector AB
uuur
’ or ‘vector ar ’ The point A from where the vector AB
uuur
starts is called its initial point, and the
point B where it ends is called its terminal point |
1 | 5035-5038 | Notice that a directed line segment is a vector (Fig 10 1(iii)), denoted as AB
uuur
or
simply as ar , and read as ‘vector AB
uuur
’ or ‘vector ar ’ The point A from where the vector AB
uuur
starts is called its initial point, and the
point B where it ends is called its terminal point The distance between initial and
terminal points of a vector is called the magnitude (or length) of the vector, denoted as
| AB
uuur
|, or | ar |, or a |
1 | 5036-5039 | 1(iii)), denoted as AB
uuur
or
simply as ar , and read as ‘vector AB
uuur
’ or ‘vector ar ’ The point A from where the vector AB
uuur
starts is called its initial point, and the
point B where it ends is called its terminal point The distance between initial and
terminal points of a vector is called the magnitude (or length) of the vector, denoted as
| AB
uuur
|, or | ar |, or a The arrow indicates the direction of the vector |
1 | 5037-5040 | The point A from where the vector AB
uuur
starts is called its initial point, and the
point B where it ends is called its terminal point The distance between initial and
terminal points of a vector is called the magnitude (or length) of the vector, denoted as
| AB
uuur
|, or | ar |, or a The arrow indicates the direction of the vector �Note Since the length is never negative, the notation | ar | < 0 has no meaning |
1 | 5038-5041 | The distance between initial and
terminal points of a vector is called the magnitude (or length) of the vector, denoted as
| AB
uuur
|, or | ar |, or a The arrow indicates the direction of the vector �Note Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector
From Class XI, recall the three dimensional right handed rectangular coordinate
system (Fig 10 |
1 | 5039-5042 | The arrow indicates the direction of the vector �Note Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector
From Class XI, recall the three dimensional right handed rectangular coordinate
system (Fig 10 2(i)) |
1 | 5040-5043 | �Note Since the length is never negative, the notation | ar | < 0 has no meaning Position Vector
From Class XI, recall the three dimensional right handed rectangular coordinate
system (Fig 10 2(i)) Consider a point P in space, having coordinates (x, y, z) with
respect to the origin O(0, 0, 0) |
1 | 5041-5044 | Position Vector
From Class XI, recall the three dimensional right handed rectangular coordinate
system (Fig 10 2(i)) Consider a point P in space, having coordinates (x, y, z) with
respect to the origin O(0, 0, 0) Then, the vector OP
uuur having O and P as its initial and
terminal points, respectively, is called the position vector of the point P with respect
to O |
1 | 5042-5045 | 2(i)) Consider a point P in space, having coordinates (x, y, z) with
respect to the origin O(0, 0, 0) Then, the vector OP
uuur having O and P as its initial and
terminal points, respectively, is called the position vector of the point P with respect
to O Using distance formula (from Class XI), the magnitude of OP
uuur (or rr ) is given by
| OP |
uuur =
2
2
2
x
y
z
+
+
In practice, the position vectors of points A, B, C, etc |
1 | 5043-5046 | Consider a point P in space, having coordinates (x, y, z) with
respect to the origin O(0, 0, 0) Then, the vector OP
uuur having O and P as its initial and
terminal points, respectively, is called the position vector of the point P with respect
to O Using distance formula (from Class XI), the magnitude of OP
uuur (or rr ) is given by
| OP |
uuur =
2
2
2
x
y
z
+
+
In practice, the position vectors of points A, B, C, etc , with respect to the origin O
are denoted by ar , ,
b c
r r , etc |
1 | 5044-5047 | Then, the vector OP
uuur having O and P as its initial and
terminal points, respectively, is called the position vector of the point P with respect
to O Using distance formula (from Class XI), the magnitude of OP
uuur (or rr ) is given by
| OP |
uuur =
2
2
2
x
y
z
+
+
In practice, the position vectors of points A, B, C, etc , with respect to the origin O
are denoted by ar , ,
b c
r r , etc , respectively (Fig 10 |
1 | 5045-5048 | Using distance formula (from Class XI), the magnitude of OP
uuur (or rr ) is given by
| OP |
uuur =
2
2
2
x
y
z
+
+
In practice, the position vectors of points A, B, C, etc , with respect to the origin O
are denoted by ar , ,
b c
r r , etc , respectively (Fig 10 2 (ii)) |
1 | 5046-5049 | , with respect to the origin O
are denoted by ar , ,
b c
r r , etc , respectively (Fig 10 2 (ii)) Fig 10 |
1 | 5047-5050 | , respectively (Fig 10 2 (ii)) Fig 10 1
© NCERT
not to be republished
MATHEMATICS
426
A
O
P
90°
X
Y
Z
X
A
O
B
P(
)
x,y,z
C
P(
x,y,z)
r
x
y
z
Direction Cosines
Consider the position vector
OP or
uuur
rr of a point P(x, y, z) as in Fig 10 |
1 | 5048-5051 | 2 (ii)) Fig 10 1
© NCERT
not to be republished
MATHEMATICS
426
A
O
P
90°
X
Y
Z
X
A
O
B
P(
)
x,y,z
C
P(
x,y,z)
r
x
y
z
Direction Cosines
Consider the position vector
OP or
uuur
rr of a point P(x, y, z) as in Fig 10 3 |
1 | 5049-5052 | Fig 10 1
© NCERT
not to be republished
MATHEMATICS
426
A
O
P
90°
X
Y
Z
X
A
O
B
P(
)
x,y,z
C
P(
x,y,z)
r
x
y
z
Direction Cosines
Consider the position vector
OP or
uuur
rr of a point P(x, y, z) as in Fig 10 3 The angles α,
β, γ made by the vector rr with the positive directions of x, y and z-axes respectively,
are called its direction angles |
1 | 5050-5053 | 1
© NCERT
not to be republished
MATHEMATICS
426
A
O
P
90°
X
Y
Z
X
A
O
B
P(
)
x,y,z
C
P(
x,y,z)
r
x
y
z
Direction Cosines
Consider the position vector
OP or
uuur
rr of a point P(x, y, z) as in Fig 10 3 The angles α,
β, γ made by the vector rr with the positive directions of x, y and z-axes respectively,
are called its direction angles The cosine values of these angles, i |
1 | 5051-5054 | 3 The angles α,
β, γ made by the vector rr with the positive directions of x, y and z-axes respectively,
are called its direction angles The cosine values of these angles, i e |
1 | 5052-5055 | The angles α,
β, γ made by the vector rr with the positive directions of x, y and z-axes respectively,
are called its direction angles The cosine values of these angles, i e , cosα, cosβ and
cos γ are called direction cosines of the vector rr , and usually denoted by l, m and n,
respectively |
1 | 5053-5056 | The cosine values of these angles, i e , cosα, cosβ and
cos γ are called direction cosines of the vector rr , and usually denoted by l, m and n,
respectively Fig 10 |
1 | 5054-5057 | e , cosα, cosβ and
cos γ are called direction cosines of the vector rr , and usually denoted by l, m and n,
respectively Fig 10 3
From Fig 10 |
1 | 5055-5058 | , cosα, cosβ and
cos γ are called direction cosines of the vector rr , and usually denoted by l, m and n,
respectively Fig 10 3
From Fig 10 3, one may note that the triangle OAP is right angled, and in it, we
have
(
)
cos
stands for | |
x
r
r
α =r
r |
1 | 5056-5059 | Fig 10 3
From Fig 10 3, one may note that the triangle OAP is right angled, and in it, we
have
(
)
cos
stands for | |
x
r
r
α =r
r Similarly, from the right angled triangles OBP and
OCP, we may write cos
y and cos
z
r
r
β =
γ = |
1 | 5057-5060 | 3
From Fig 10 3, one may note that the triangle OAP is right angled, and in it, we
have
(
)
cos
stands for | |
x
r
r
α =r
r Similarly, from the right angled triangles OBP and
OCP, we may write cos
y and cos
z
r
r
β =
γ = Thus, the coordinates of the point P may
also be expressed as (lr, mr,nr) |
1 | 5058-5061 | 3, one may note that the triangle OAP is right angled, and in it, we
have
(
)
cos
stands for | |
x
r
r
α =r
r Similarly, from the right angled triangles OBP and
OCP, we may write cos
y and cos
z
r
r
β =
γ = Thus, the coordinates of the point P may
also be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction
cosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively |
1 | 5059-5062 | Similarly, from the right angled triangles OBP and
OCP, we may write cos
y and cos
z
r
r
β =
γ = Thus, the coordinates of the point P may
also be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction
cosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10 |
1 | 5060-5063 | Thus, the coordinates of the point P may
also be expressed as (lr, mr,nr) The numbers lr, mr and nr, proportional to the direction
cosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10 2
© NCERT
not to be republished
VECTOR ALGEBRA
427
�Note One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 ≠ 1, in general |
1 | 5061-5064 | The numbers lr, mr and nr, proportional to the direction
cosines are called as direction ratios of vector rr , and denoted as a, b and c, respectively Fig 10 2
© NCERT
not to be republished
VECTOR ALGEBRA
427
�Note One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 ≠ 1, in general 10 |
1 | 5062-5065 | Fig 10 2
© NCERT
not to be republished
VECTOR ALGEBRA
427
�Note One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 ≠ 1, in general 10 3 Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero
vector (or null vector), and denoted as 0
r |
1 | 5063-5066 | 2
© NCERT
not to be republished
VECTOR ALGEBRA
427
�Note One may note that l2 + m2 + n2 = 1 but a2 + b2 + c2 ≠ 1, in general 10 3 Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero
vector (or null vector), and denoted as 0
r Zero vector can not be assigned a definite
direction as it has zero magnitude |
1 | 5064-5067 | 10 3 Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero
vector (or null vector), and denoted as 0
r Zero vector can not be assigned a definite
direction as it has zero magnitude Or, alternatively otherwise, it may be regarded as
having any direction |
1 | 5065-5068 | 3 Types of Vectors
Zero Vector A vector whose initial and terminal points coincide, is called a zero
vector (or null vector), and denoted as 0
r Zero vector can not be assigned a definite
direction as it has zero magnitude Or, alternatively otherwise, it may be regarded as
having any direction The vectors AA, BB
uuur uuur
represent the zero vector,
Unit Vector A vector whose magnitude is unity (i |
1 | 5066-5069 | Zero vector can not be assigned a definite
direction as it has zero magnitude Or, alternatively otherwise, it may be regarded as
having any direction The vectors AA, BB
uuur uuur
represent the zero vector,
Unit Vector A vector whose magnitude is unity (i e |
1 | 5067-5070 | Or, alternatively otherwise, it may be regarded as
having any direction The vectors AA, BB
uuur uuur
represent the zero vector,
Unit Vector A vector whose magnitude is unity (i e , 1 unit) is called a unit vector |
1 | 5068-5071 | The vectors AA, BB
uuur uuur
represent the zero vector,
Unit Vector A vector whose magnitude is unity (i e , 1 unit) is called a unit vector The
unit vector in the direction of a given vector ar is denoted by ˆa |
1 | 5069-5072 | e , 1 unit) is called a unit vector The
unit vector in the direction of a given vector ar is denoted by ˆa Coinitial Vectors Two or more vectors having the same initial point are called coinitial
vectors |
1 | 5070-5073 | , 1 unit) is called a unit vector The
unit vector in the direction of a given vector ar is denoted by ˆa Coinitial Vectors Two or more vectors having the same initial point are called coinitial
vectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to
the same line, irrespective of their magnitudes and directions |
1 | 5071-5074 | The
unit vector in the direction of a given vector ar is denoted by ˆa Coinitial Vectors Two or more vectors having the same initial point are called coinitial
vectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to
the same line, irrespective of their magnitudes and directions Equal Vectors Two vectors
aand
rb
r
are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as
a =
rb
r |
1 | 5072-5075 | Coinitial Vectors Two or more vectors having the same initial point are called coinitial
vectors Collinear Vectors Two or more vectors are said to be collinear if they are parallel to
the same line, irrespective of their magnitudes and directions Equal Vectors Two vectors
aand
rb
r
are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as
a =
rb
r Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, AB
uuur ), but direction is opposite to that of it, is called negative of the given vector |
1 | 5073-5076 | Collinear Vectors Two or more vectors are said to be collinear if they are parallel to
the same line, irrespective of their magnitudes and directions Equal Vectors Two vectors
aand
rb
r
are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as
a =
rb
r Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, AB
uuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA
uuur is negative of the vector AB
uuur , and written as BA
uuur= −AB
uuur |
1 | 5074-5077 | Equal Vectors Two vectors
aand
rb
r
are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as
a =
rb
r Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, AB
uuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA
uuur is negative of the vector AB
uuur , and written as BA
uuur= −AB
uuur Remark The vectors defined above are such that any of them may be subject to its
parallel displacement without changing its magnitude and direction |
1 | 5075-5078 | Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, AB
uuur ), but direction is opposite to that of it, is called negative of the given vector For example, vector BA
uuur is negative of the vector AB
uuur , and written as BA
uuur= −AB
uuur Remark The vectors defined above are such that any of them may be subject to its
parallel displacement without changing its magnitude and direction Such vectors are
called free vectors |
1 | 5076-5079 | For example, vector BA
uuur is negative of the vector AB
uuur , and written as BA
uuur= −AB
uuur Remark The vectors defined above are such that any of them may be subject to its
parallel displacement without changing its magnitude and direction Such vectors are
called free vectors Throughout this chapter, we will be dealing with free vectors only |
1 | 5077-5080 | Remark The vectors defined above are such that any of them may be subject to its
parallel displacement without changing its magnitude and direction Such vectors are
called free vectors Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement
of 40 km, 30° west of south |
1 | 5078-5081 | Such vectors are
called free vectors Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement
of 40 km, 30° west of south Solution The vector OP
uuur represents the required
displacement (Fig 10 |
1 | 5079-5082 | Throughout this chapter, we will be dealing with free vectors only Example 1 Represent graphically a displacement
of 40 km, 30° west of south Solution The vector OP
uuur represents the required
displacement (Fig 10 4) |
1 | 5080-5083 | Example 1 Represent graphically a displacement
of 40 km, 30° west of south Solution The vector OP
uuur represents the required
displacement (Fig 10 4) Example 2 Classify the following measures as
scalars and vectors |
1 | 5081-5084 | Solution The vector OP
uuur represents the required
displacement (Fig 10 4) Example 2 Classify the following measures as
scalars and vectors (i) 5 seconds
(ii) 1000 cm3
Fig 10 |
1 | 5082-5085 | 4) Example 2 Classify the following measures as
scalars and vectors (i) 5 seconds
(ii) 1000 cm3
Fig 10 4
© NCERT
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MATHEMATICS
428
Fig 10 |
1 | 5083-5086 | Example 2 Classify the following measures as
scalars and vectors (i) 5 seconds
(ii) 1000 cm3
Fig 10 4
© NCERT
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MATHEMATICS
428
Fig 10 5
(iii) 10 Newton
(iv) 30 km/hr
(v) 10 g/cm3
(vi) 20 m/s towards north
Solution
(i) Time-scalar
(ii) Volume-scalar
(iii) Force-vector
(iv) Speed-scalar
(v) Density-scalar
(vi) Velocity-vector
Example 3 In Fig 10 |
1 | 5084-5087 | (i) 5 seconds
(ii) 1000 cm3
Fig 10 4
© NCERT
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MATHEMATICS
428
Fig 10 5
(iii) 10 Newton
(iv) 30 km/hr
(v) 10 g/cm3
(vi) 20 m/s towards north
Solution
(i) Time-scalar
(ii) Volume-scalar
(iii) Force-vector
(iv) Speed-scalar
(v) Density-scalar
(vi) Velocity-vector
Example 3 In Fig 10 5, which of the vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
Solution
(i) Collinear vectors : ,
a cand
rd
r
r |
1 | 5085-5088 | 4
© NCERT
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MATHEMATICS
428
Fig 10 5
(iii) 10 Newton
(iv) 30 km/hr
(v) 10 g/cm3
(vi) 20 m/s towards north
Solution
(i) Time-scalar
(ii) Volume-scalar
(iii) Force-vector
(iv) Speed-scalar
(v) Density-scalar
(vi) Velocity-vector
Example 3 In Fig 10 5, which of the vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
Solution
(i) Collinear vectors : ,
a cand
rd
r
r (ii) Equal vectors :
and |
1 | 5086-5089 | 5
(iii) 10 Newton
(iv) 30 km/hr
(v) 10 g/cm3
(vi) 20 m/s towards north
Solution
(i) Time-scalar
(ii) Volume-scalar
(iii) Force-vector
(iv) Speed-scalar
(v) Density-scalar
(vi) Velocity-vector
Example 3 In Fig 10 5, which of the vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
Solution
(i) Collinear vectors : ,
a cand
rd
r
r (ii) Equal vectors :
and a
c
r
r
(iii) Coinitial vectors : ,
and |
1 | 5087-5090 | 5, which of the vectors are:
(i) Collinear
(ii) Equal
(iii) Coinitial
Solution
(i) Collinear vectors : ,
a cand
rd
r
r (ii) Equal vectors :
and a
c
r
r
(iii) Coinitial vectors : ,
and b c
d
r
r
r
EXERCISE 10 |
1 | 5088-5091 | (ii) Equal vectors :
and a
c
r
r
(iii) Coinitial vectors : ,
and b c
d
r
r
r
EXERCISE 10 1
1 |
1 | 5089-5092 | a
c
r
r
(iii) Coinitial vectors : ,
and b c
d
r
r
r
EXERCISE 10 1
1 Represent graphically a displacement of 40 km, 30° east of north |
1 | 5090-5093 | b c
d
r
r
r
EXERCISE 10 1
1 Represent graphically a displacement of 40 km, 30° east of north 2 |
1 | 5091-5094 | 1
1 Represent graphically a displacement of 40 km, 30° east of north 2 Classify the following measures as scalars and vectors |
1 | 5092-5095 | Represent graphically a displacement of 40 km, 30° east of north 2 Classify the following measures as scalars and vectors (i) 10 kg
(ii) 2 meters north-west
(iii) 40°
(iv) 40 watt
(v) 10–19 coulomb
(vi) 20 m/s2
3 |
1 | 5093-5096 | 2 Classify the following measures as scalars and vectors (i) 10 kg
(ii) 2 meters north-west
(iii) 40°
(iv) 40 watt
(v) 10–19 coulomb
(vi) 20 m/s2
3 Classify the following as scalar and vector quantities |
1 | 5094-5097 | Classify the following measures as scalars and vectors (i) 10 kg
(ii) 2 meters north-west
(iii) 40°
(iv) 40 watt
(v) 10–19 coulomb
(vi) 20 m/s2
3 Classify the following as scalar and vector quantities (i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done
4 |
1 | 5095-5098 | (i) 10 kg
(ii) 2 meters north-west
(iii) 40°
(iv) 40 watt
(v) 10–19 coulomb
(vi) 20 m/s2
3 Classify the following as scalar and vector quantities (i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done
4 In Fig 10 |
1 | 5096-5099 | Classify the following as scalar and vector quantities (i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done
4 In Fig 10 6 (a square), identify the following vectors |
1 | 5097-5100 | (i) time period
(ii) distance
(iii) force
(iv) velocity
(v) work done
4 In Fig 10 6 (a square), identify the following vectors (i) Coinitial
(ii) Equal
(iii) Collinear but not equal
5 |
1 | 5098-5101 | In Fig 10 6 (a square), identify the following vectors (i) Coinitial
(ii) Equal
(iii) Collinear but not equal
5 Answer the following as true or false |
1 | 5099-5102 | 6 (a square), identify the following vectors (i) Coinitial
(ii) Equal
(iii) Collinear but not equal
5 Answer the following as true or false (i) ar and
a
− r are collinear |
1 | 5100-5103 | (i) Coinitial
(ii) Equal
(iii) Collinear but not equal
5 Answer the following as true or false (i) ar and
a
− r are collinear (ii) Two collinear vectors are always equal in
magnitude |
1 | 5101-5104 | Answer the following as true or false (i) ar and
a
− r are collinear (ii) Two collinear vectors are always equal in
magnitude (iii) Two vectors having same magnitude are collinear |
1 | 5102-5105 | (i) ar and
a
− r are collinear (ii) Two collinear vectors are always equal in
magnitude (iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal |
1 | 5103-5106 | (ii) Two collinear vectors are always equal in
magnitude (iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal Fig 10 |
1 | 5104-5107 | (iii) Two vectors having same magnitude are collinear (iv) Two collinear vectors having the same magnitude are equal Fig 10 6
© NCERT
not to be republished
VECTOR ALGEBRA
429
10 |
1 | 5105-5108 | (iv) Two collinear vectors having the same magnitude are equal Fig 10 6
© NCERT
not to be republished
VECTOR ALGEBRA
429
10 4 Addition of Vectors
A vector AB
uuur
simply means the displacement from a
point A to the point B |
1 | 5106-5109 | Fig 10 6
© NCERT
not to be republished
VECTOR ALGEBRA
429
10 4 Addition of Vectors
A vector AB
uuur
simply means the displacement from a
point A to the point B Now consider a situation that a
girl moves from A to B and then from B to C
(Fig 10 |
1 | 5107-5110 | 6
© NCERT
not to be republished
VECTOR ALGEBRA
429
10 4 Addition of Vectors
A vector AB
uuur
simply means the displacement from a
point A to the point B Now consider a situation that a
girl moves from A to B and then from B to C
(Fig 10 7) |
1 | 5108-5111 | 4 Addition of Vectors
A vector AB
uuur
simply means the displacement from a
point A to the point B Now consider a situation that a
girl moves from A to B and then from B to C
(Fig 10 7) The net displacement made by the girl from
point A to the point C, is given by the vector AC
uuur and
expressed as
AC
uuur = AB
BC
+
uuur
uuur
This is known as the triangle law of vector addition |
1 | 5109-5112 | Now consider a situation that a
girl moves from A to B and then from B to C
(Fig 10 7) The net displacement made by the girl from
point A to the point C, is given by the vector AC
uuur and
expressed as
AC
uuur = AB
BC
+
uuur
uuur
This is known as the triangle law of vector addition In general, if we have two vectors ar and b
r
(Fig 10 |
1 | 5110-5113 | 7) The net displacement made by the girl from
point A to the point C, is given by the vector AC
uuur and
expressed as
AC
uuur = AB
BC
+
uuur
uuur
This is known as the triangle law of vector addition In general, if we have two vectors ar and b
r
(Fig 10 8 (i)), then to add them, they
are positioned so that the initial point of one coincides with the terminal point of the
other (Fig 10 |
1 | 5111-5114 | The net displacement made by the girl from
point A to the point C, is given by the vector AC
uuur and
expressed as
AC
uuur = AB
BC
+
uuur
uuur
This is known as the triangle law of vector addition In general, if we have two vectors ar and b
r
(Fig 10 8 (i)), then to add them, they
are positioned so that the initial point of one coincides with the terminal point of the
other (Fig 10 8(ii)) |
1 | 5112-5115 | In general, if we have two vectors ar and b
r
(Fig 10 8 (i)), then to add them, they
are positioned so that the initial point of one coincides with the terminal point of the
other (Fig 10 8(ii)) Fig 10 |
1 | 5113-5116 | 8 (i)), then to add them, they
are positioned so that the initial point of one coincides with the terminal point of the
other (Fig 10 8(ii)) Fig 10 8
For example, in Fig 10 |
1 | 5114-5117 | 8(ii)) Fig 10 8
For example, in Fig 10 8 (ii), we have shifted vector b
r without changing its magnitude
and direction, so that it’s initial point coincides with the terminal point of ar |
1 | 5115-5118 | Fig 10 8
For example, in Fig 10 8 (ii), we have shifted vector b
r without changing its magnitude
and direction, so that it’s initial point coincides with the terminal point of ar Then, the
vector a
+b
r
r
, represented by the third side AC of the triangle ABC, gives us the sum
(or resultant) of the vectors ar and b
r
i |
1 | 5116-5119 | 8
For example, in Fig 10 8 (ii), we have shifted vector b
r without changing its magnitude
and direction, so that it’s initial point coincides with the terminal point of ar Then, the
vector a
+b
r
r
, represented by the third side AC of the triangle ABC, gives us the sum
(or resultant) of the vectors ar and b
r
i e |
1 | 5117-5120 | 8 (ii), we have shifted vector b
r without changing its magnitude
and direction, so that it’s initial point coincides with the terminal point of ar Then, the
vector a
+b
r
r
, represented by the third side AC of the triangle ABC, gives us the sum
(or resultant) of the vectors ar and b
r
i e , in triangle ABC (Fig 10 |
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