Vectors
A vector has direction and magnitude both but scalar has only
magnitude. e.g. Vector quantities are displacement, velocity,
acceleration, etc. and scalar quantities are length, mass, time, etc.
Characteristics of a Vector
(i) Magnitude The length of the vector AB or a is called the
magnitude of AB or a and it is represented as AB or a .
(ii) Sense The direction of a line segment from its initial point to
its terminal point is called its sense.
e.g. The sense of AB is from A to B and that of BA is from B to A.
initial point Terminal point
A B
(iii) Support The line of infinite length of which the line segment
AB is a part, is called the support of the vector AB.
A B
Support
Types of Vectors
(i) Zero or Null Vector A vector whose initial and terminal
points are coincident is called zero or null vector. It is denoted
by 0.
(ii) Unit Vector A vector whose magnitude is unity i.e., 1 unit is
called a unit vector. The unit vector in the direction of n is given
n
by and it is denoted by n.
$
|n|
(iii) Free Vector If the initial point of a vector is not specified, then
it is said to be a free vector.
(iv) Like and Unlike Vectors Vectors are said to be like when
they have the same direction and unlike when they have
opposite direction.
(v) Collinear or Parallel Vectors Vectors having the same or
parallel supports are called collinear vectors.
, (vi) Equal Vectors Two vectors a and b are said to be equal, written
as a = b, if they have same length and same direction.
(vii) Negative Vector A vector having the same magnitude as that
of a given vector a and the direction opposite to that of a is called
the negative vector a and it is denoted by − a.
(viii) Coinitial Vectors Vectors having same initial point are
called coinitial vectors.
(ix) Coterminus Vectors Vectors having the same terminal point
are called coterminus vectors.
(x) Localised Vectors A vector which is drawn parallel to a given
vector through a specified point in space is called localised
vector.
(xi) Coplanar Vectors A system of vectors is said to be coplanar,
if their supports are parallel to the same plane. Otherwise they
are called non-coplanar vectors.
(xii) Reciprocal of a Vector A vector having the same direction as
that of a given vector but magnitude equal to the reciprocal of
the given vector is known as the reciprocal of a and it is denoted
1
by a −1, i.e. if|a| = a , then|a−1| = .
a
Addition of Vectors
Triangle Law of Vector Addition
Let a and b be any two vectors. From the terminal point of a, vector b
is drawn. Then, the vector from the initial point O of a to the terminal
point B of b is called the sum of vectors a and b and is denoted by
a + b. This is called the triangle law of addition of vectors.
B
a+b
b
O A
a
Note When the sides of a triangle are taken in order, then the resultant will be
AB + BC + CA = 0
A vector has direction and magnitude both but scalar has only
magnitude. e.g. Vector quantities are displacement, velocity,
acceleration, etc. and scalar quantities are length, mass, time, etc.
Characteristics of a Vector
(i) Magnitude The length of the vector AB or a is called the
magnitude of AB or a and it is represented as AB or a .
(ii) Sense The direction of a line segment from its initial point to
its terminal point is called its sense.
e.g. The sense of AB is from A to B and that of BA is from B to A.
initial point Terminal point
A B
(iii) Support The line of infinite length of which the line segment
AB is a part, is called the support of the vector AB.
A B
Support
Types of Vectors
(i) Zero or Null Vector A vector whose initial and terminal
points are coincident is called zero or null vector. It is denoted
by 0.
(ii) Unit Vector A vector whose magnitude is unity i.e., 1 unit is
called a unit vector. The unit vector in the direction of n is given
n
by and it is denoted by n.
$
|n|
(iii) Free Vector If the initial point of a vector is not specified, then
it is said to be a free vector.
(iv) Like and Unlike Vectors Vectors are said to be like when
they have the same direction and unlike when they have
opposite direction.
(v) Collinear or Parallel Vectors Vectors having the same or
parallel supports are called collinear vectors.
, (vi) Equal Vectors Two vectors a and b are said to be equal, written
as a = b, if they have same length and same direction.
(vii) Negative Vector A vector having the same magnitude as that
of a given vector a and the direction opposite to that of a is called
the negative vector a and it is denoted by − a.
(viii) Coinitial Vectors Vectors having same initial point are
called coinitial vectors.
(ix) Coterminus Vectors Vectors having the same terminal point
are called coterminus vectors.
(x) Localised Vectors A vector which is drawn parallel to a given
vector through a specified point in space is called localised
vector.
(xi) Coplanar Vectors A system of vectors is said to be coplanar,
if their supports are parallel to the same plane. Otherwise they
are called non-coplanar vectors.
(xii) Reciprocal of a Vector A vector having the same direction as
that of a given vector but magnitude equal to the reciprocal of
the given vector is known as the reciprocal of a and it is denoted
1
by a −1, i.e. if|a| = a , then|a−1| = .
a
Addition of Vectors
Triangle Law of Vector Addition
Let a and b be any two vectors. From the terminal point of a, vector b
is drawn. Then, the vector from the initial point O of a to the terminal
point B of b is called the sum of vectors a and b and is denoted by
a + b. This is called the triangle law of addition of vectors.
B
a+b
b
O A
a
Note When the sides of a triangle are taken in order, then the resultant will be
AB + BC + CA = 0
