Page 1
MATHEMATICS 338
vIn most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL v
10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is a real number. Such quantities are called scalars.
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.
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. 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.R. Hamilton
(1805-1865)
Rationalised 2023-24
Page 2
MATHEMATICS 338
vIn most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL v
10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is a real number. Such quantities are called scalars.
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.
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. 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.R. Hamilton
(1805-1865)
Rationalised 2023-24
VECTOR ALGEBRA 339
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.
Notice that a directed line segment is a vector (Fig 10.1(iii)), denoted as or
simply as , and read as ‘vector ’ or ‘vector ’.
The point A from where the vector 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
| |, or | |, or a. The arrow indicates the direction of the vector.
A
Note Since the length is never negative, the notation | | < 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). Then, the vector 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 (or ) is given by
| |=
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 , , , etc., respectively (Fig 10.2 (ii)).
Fig 10.1
Rationalised 2023-24
Page 3
MATHEMATICS 338
vIn most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL v
10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is a real number. Such quantities are called scalars.
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.
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. 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.R. Hamilton
(1805-1865)
Rationalised 2023-24
VECTOR ALGEBRA 339
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.
Notice that a directed line segment is a vector (Fig 10.1(iii)), denoted as or
simply as , and read as ‘vector ’ or ‘vector ’.
The point A from where the vector 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
| |, or | |, or a. The arrow indicates the direction of the vector.
A
Note Since the length is never negative, the notation | | < 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). Then, the vector 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 (or ) is given by
| |=
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 , , , etc., respectively (Fig 10.2 (ii)).
Fig 10.1
Rationalised 2023-24
MATHEMATICS 340
A
O
P
a 90°
X
Y
Z
X
A
O
B
P( ) x,y,z
C
a b g P( ) x,y,z
r
x
y
z
Direction Cosines
Consider the position vector of a point P(x, y, z) as in Fig 10.3. The angles a,
ß, ? made by the vector with the positive directions of x, y and z-axes respectively,
are called its direction angles. The cosine values of these angles, i.e., cos a, cosß and
cos ? are called direction cosines of the vector , 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 . Similarly, from the right angled triangles OBP and
OCP , we may write cos and cos
y z
r r
ß = ? = . 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 , and denoted as a, b and c, respectively.
Fig 10.2
Rationalised 2023-24
Page 4
MATHEMATICS 338
vIn most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL v
10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is a real number. Such quantities are called scalars.
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.
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. 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.R. Hamilton
(1805-1865)
Rationalised 2023-24
VECTOR ALGEBRA 339
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.
Notice that a directed line segment is a vector (Fig 10.1(iii)), denoted as or
simply as , and read as ‘vector ’ or ‘vector ’.
The point A from where the vector 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
| |, or | |, or a. The arrow indicates the direction of the vector.
A
Note Since the length is never negative, the notation | | < 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). Then, the vector 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 (or ) is given by
| |=
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 , , , etc., respectively (Fig 10.2 (ii)).
Fig 10.1
Rationalised 2023-24
MATHEMATICS 340
A
O
P
a 90°
X
Y
Z
X
A
O
B
P( ) x,y,z
C
a b g P( ) x,y,z
r
x
y
z
Direction Cosines
Consider the position vector of a point P(x, y, z) as in Fig 10.3. The angles a,
ß, ? made by the vector with the positive directions of x, y and z-axes respectively,
are called its direction angles. The cosine values of these angles, i.e., cos a, cosß and
cos ? are called direction cosines of the vector , 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 . Similarly, from the right angled triangles OBP and
OCP , we may write cos and cos
y z
r r
ß = ? = . 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 , and denoted as a, b and c, respectively.
Fig 10.2
Rationalised 2023-24
VECTOR ALGEBRA 341
A
Note One may note that l
2
+ m
2
+ n
2
= 1 but a
2
+ b
2
+ c
2
? 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 . 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 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 is denoted by ˆ a .
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 are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as .
Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, ), but direction is opposite to that of it, is called negative of the given vector.
For example, vector is negative of the vector , and written as = – .
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.
Solution The vector represents the required
displacement (Fig 10.4).
Example 2 Classify the following measures as
scalars and vectors.
(i) 5 seconds
(ii) 1000 cm
3
Fig 10.4
Rationalised 2023-24
Page 5
MATHEMATICS 338
vIn most sciences one generation tears down what another has built and what
one has established another undoes. In Mathematics alone each generation
builds a new story to the old structure. – HERMAN HANKEL v
10.1 Introduction
In our day to day life, we come across many queries such
as – What is your height? How should a football player hit
the ball to give a pass to another player of his team? Observe
that a possible answer to the first query may be 1.6 meters,
a quantity that involves only one value (magnitude) which
is a real number. Such quantities are called scalars.
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.
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. 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.R. Hamilton
(1805-1865)
Rationalised 2023-24
VECTOR ALGEBRA 339
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.
Notice that a directed line segment is a vector (Fig 10.1(iii)), denoted as or
simply as , and read as ‘vector ’ or ‘vector ’.
The point A from where the vector 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
| |, or | |, or a. The arrow indicates the direction of the vector.
A
Note Since the length is never negative, the notation | | < 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). Then, the vector 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 (or ) is given by
| |=
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 , , , etc., respectively (Fig 10.2 (ii)).
Fig 10.1
Rationalised 2023-24
MATHEMATICS 340
A
O
P
a 90°
X
Y
Z
X
A
O
B
P( ) x,y,z
C
a b g P( ) x,y,z
r
x
y
z
Direction Cosines
Consider the position vector of a point P(x, y, z) as in Fig 10.3. The angles a,
ß, ? made by the vector with the positive directions of x, y and z-axes respectively,
are called its direction angles. The cosine values of these angles, i.e., cos a, cosß and
cos ? are called direction cosines of the vector , 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 . Similarly, from the right angled triangles OBP and
OCP , we may write cos and cos
y z
r r
ß = ? = . 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 , and denoted as a, b and c, respectively.
Fig 10.2
Rationalised 2023-24
VECTOR ALGEBRA 341
A
Note One may note that l
2
+ m
2
+ n
2
= 1 but a
2
+ b
2
+ c
2
? 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 . 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 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 is denoted by ˆ a .
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 are said to be equal, if they have the same
magnitude and direction regardless of the positions of their initial points, and written
as .
Negative of a Vector A vector whose magnitude is the same as that of a given vector
(say, ), but direction is opposite to that of it, is called negative of the given vector.
For example, vector is negative of the vector , and written as = – .
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.
Solution The vector represents the required
displacement (Fig 10.4).
Example 2 Classify the following measures as
scalars and vectors.
(i) 5 seconds
(ii) 1000 cm
3
Fig 10.4
Rationalised 2023-24
MATHEMATICS 342
Fig 10.5
(iii) 10 Newton (iv) 30 km/hr (v) 10 g/cm
3
(vi) 20 m/s towards north
Solution
(i) Time-scalar (ii) V olume-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 : .
(ii) Equal vectors :
(iii) Coinitial vectors :
EXERCISE 10.1
1. 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/s
2
3. Classify the following as scalar and vector quantities.
(i) time period (ii) distance (iii) force
(iv) velocity (v) work done
4. In Fig 10.6 (a square), identify the following vectors.
(i) Coinitial (ii) Equal
(iii) Collinear but not equal
5. Answer the following as true or false.
(i) and – 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.
Fig 10.6
Rationalised 2023-24
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