VECTOR ALGEBRA
Vector and Scalar
(i) Scalar quantity is a quantity having magnitude but no direction. e.g. work,volume,
time, mass, length, etc.
(ii) Vector quantity is a quantity having both direction and magnitude. e.g. displacement,
force, velocity, momentum, etc.
Representation of Vectors
A vector is often denoted by two letters with an arrow over it, i.e. ⃗⃗⃗⃗⃗
𝐀𝐁. A is called the
origin and B the terminus. Its magnitude is given by the length AB and direction is from A
to B as indicated by the arrow, we write vector quantities also in single letter notation
like a, b, c and the corresponding letters a, b, c shows their magnitude.
A B
Thus, if ⃗⃗⃗⃗⃗
𝐀𝐁 = a, then |𝑨𝑩| means the magnitude of vector a.
Types of Vectors
1. Zero or Null vector :
A vector, whose initial and terminal points coincides is called zero or null vector. Thus,
the modulus of the null vector is zero and it is denoted by 0 or ⃗𝟎. Vectors other than the
null vector are called proper vectors.
2. Unit vector:
A vector, whose magnitude is of unit length is called a unit vector. If a is a vector whose
magnitude is a, then unit vector in the direction of a is denoted by 𝐚̂ and is obtained by
dividing the vector a by its magnitude |𝐚̂|.
𝐚
Thus, 𝐚̂ = |𝐚|
,3. Coinitial vectors:
Vectors having the same initial point are called coinitial vectors.
4. Collinear or parallel vectors:
The vectors which are parallel to the same straight line.
5. Coplanar vectors:
Three or more vectors are said to be coplanar when they are parallel to the same plane
otherwise they are said to be non-coplanar vector whatever their magnitudes be.
6. Coterminous vectors:
Vectors having the same terminal points are called coterminous vectors.
7. Negative of a vector:
The vector which has the same magnitude as the vector a but opposite in direction, is
called the negative of a and is denoted by –a.
8. Reciprocal of a vector:
A vector having the same direction as that of a given vector a but magnitude is equal to
the reciprocal of the given vector, a and is denoted by a-1 .
𝟏
If |𝐚| = a, then |𝐚−𝟏 | =
𝐚
Addition of Vectors
The addition of two vectors a and b is denoted by a+b and it is known as resultant of a
and b.
There are three methods of addition of vectors
, 1. Triangle Law
R
c=a+b
b
P
a Q
[T
y
If a and b lies palong two consecutive sides of a triangle, then third side represents the
sum a+b. Symbolically,
e we have PQ+QR=PR
a or a+b= c
q
u
ot R
2. ParallelogrameLaw Q
b
fr c=a+b
o
m
th O a P
e
If two vectors da and b are represented in magnitude and direction by the two adjacent
sides of a parallelogram,
oc then their sum c is represented by the diagonal of the
parallelogram,uwhich is coinitial with the given vectors.
Symbolically, mwe have OP+OR=OQ or a+b=c
e
nt
3. Polygon Law of or Addition of Vectors
th
e
su
m
m
ar
y
of
a
n
in
Vector and Scalar
(i) Scalar quantity is a quantity having magnitude but no direction. e.g. work,volume,
time, mass, length, etc.
(ii) Vector quantity is a quantity having both direction and magnitude. e.g. displacement,
force, velocity, momentum, etc.
Representation of Vectors
A vector is often denoted by two letters with an arrow over it, i.e. ⃗⃗⃗⃗⃗
𝐀𝐁. A is called the
origin and B the terminus. Its magnitude is given by the length AB and direction is from A
to B as indicated by the arrow, we write vector quantities also in single letter notation
like a, b, c and the corresponding letters a, b, c shows their magnitude.
A B
Thus, if ⃗⃗⃗⃗⃗
𝐀𝐁 = a, then |𝑨𝑩| means the magnitude of vector a.
Types of Vectors
1. Zero or Null vector :
A vector, whose initial and terminal points coincides is called zero or null vector. Thus,
the modulus of the null vector is zero and it is denoted by 0 or ⃗𝟎. Vectors other than the
null vector are called proper vectors.
2. Unit vector:
A vector, whose magnitude is of unit length is called a unit vector. If a is a vector whose
magnitude is a, then unit vector in the direction of a is denoted by 𝐚̂ and is obtained by
dividing the vector a by its magnitude |𝐚̂|.
𝐚
Thus, 𝐚̂ = |𝐚|
,3. Coinitial vectors:
Vectors having the same initial point are called coinitial vectors.
4. Collinear or parallel vectors:
The vectors which are parallel to the same straight line.
5. Coplanar vectors:
Three or more vectors are said to be coplanar when they are parallel to the same plane
otherwise they are said to be non-coplanar vector whatever their magnitudes be.
6. Coterminous vectors:
Vectors having the same terminal points are called coterminous vectors.
7. Negative of a vector:
The vector which has the same magnitude as the vector a but opposite in direction, is
called the negative of a and is denoted by –a.
8. Reciprocal of a vector:
A vector having the same direction as that of a given vector a but magnitude is equal to
the reciprocal of the given vector, a and is denoted by a-1 .
𝟏
If |𝐚| = a, then |𝐚−𝟏 | =
𝐚
Addition of Vectors
The addition of two vectors a and b is denoted by a+b and it is known as resultant of a
and b.
There are three methods of addition of vectors
, 1. Triangle Law
R
c=a+b
b
P
a Q
[T
y
If a and b lies palong two consecutive sides of a triangle, then third side represents the
sum a+b. Symbolically,
e we have PQ+QR=PR
a or a+b= c
q
u
ot R
2. ParallelogrameLaw Q
b
fr c=a+b
o
m
th O a P
e
If two vectors da and b are represented in magnitude and direction by the two adjacent
sides of a parallelogram,
oc then their sum c is represented by the diagonal of the
parallelogram,uwhich is coinitial with the given vectors.
Symbolically, mwe have OP+OR=OQ or a+b=c
e
nt
3. Polygon Law of or Addition of Vectors
th
e
su
m
m
ar
y
of
a
n
in
