quantity possessing both magnitude and direction, and which can be represented
by a directed straight line segment, is called a vector. Obviously, a vector
related to a physical quantity will also have a unit for its magnitude. A
scalar is a quantity possessing magnitude (with a unit) only. The ratio of two
scalars does not have a unit. Mathematically, a scalar is just a real number.
Examples of vectors are displacement, velocity, acceleration, force, electric
field intensity, magnetic field intensity, etc. Examples of scalars are
distance, energy, voltage, permeability, etc. Finite angular rotations of a
body about a point possess both magnitude and direction but they are not
vectors since they cannot be represented by a directed straight line segment.
Infinitesimal rotations can be treated as vectors.
Representation of a vector:
represent a vector by a line segment having a direction. The length of the line
segment is a measure of the magnitude of the vector (with/without some suitable
scale) and the direction is indicated by putting an arrow anywhere on that line
segment. We write it as or , or simply as or
and read as vector AB or a. A is
called the initial point and B the terminal point (or terminus) of the vector.
Line AB produced on both sides is called the line of support.
magnitude of the vector is denoted by or(read as modulus of vector ). Just letter ‘a’ can also be used
to denote its magnitude. Two vectors are said to be equal if they have (i) the
same length, (ii) the same or parallel supports and (iii) the same sense.
In this chapter we
will deal with only those vectors which can be moved anywhere in space
protecting their magnitude and direction (called free vectors as against
those which are fixed in space) i.e., the vector will be assumed unchanged if it
is transferred parallel to its direction anywhere in space. Consequently two
vectors are said to be equal if they have the same magnitude and direction.
Position vector of a point
We take arbitrarily any
point O in space to be called the origin of reference. The position vector
(p.v.) of any point P, with respect to the origin is the vector . For any two points P and Q in
space, the equality = expresses
any vector in terms of the position vectors and of
P and Q respectively.
Angle between two vectors
It is defined as the
smaller angle formed when the initial points or the terminal points of two
vectors are brought together.
Note: 0° £ q £
Operations on Vectors
Multiplication of a vector
by a scalar:
Given a vector and a scalar k∈R, then k(or
k) denotes a vector whose magnitude
is i.e., k times that of and whose direction is the same or
opposite to that of according as k > 0 or k
< 0 respectively. Also, 0=, zero or null vector which has zero
magnitude and arbitrary direction.
(-1) = -,
When we have two
vectors and such
that = k, k∈R, then and
are called collinear vectors. is said to be a scalar multiple of . and
are parallel if k > 0 and anti
parallel if k < 0.
Note also that k1 (k2) = (k1k2) = k2(k1) " k1, k2 ∈ R.
A vector having the magnitude as one (unity)
is called a unit vector.
Unit vector in the direction of is defined as = and
is denoted by
(read as a cap).
Addition of Vectors:
two vectors and ,
their sum or resultant written as (+ ) is a vector obtained by first
bringing the initial point of to the terminal
point of and then joining the initial point of to the terminal point of giving a consistent direction by
completing the triangle OAB
The sum can also be obtained by bringing the
initial points of and together
and then completing the parallelogram OACB
Note that addition is commutative i.e., + =
Also, + (+) =
(+ ) +
i.e. the addition of vectors obeys the associative
law. If and are
collinear, their sum is still obtained in the same manner although we do not
have a triangle or a parallelogram in this case.
For adding more than two vectors, we have a polygon
law of addition which is just an extension of the triangle law.
A consequence of this is that, if the terminus of the
last vector coincides with the initial point of the first vector, the sum of
the vectors is .
To obtain (difference of two
vectors), perform addition ofand .
Also, = ; + = ;
(k1 + k2)= k1+ k2; k (=
vectors (non-zero and non-collinear) constitute a plane. Their sum or
difference also lies in the same plane. Three vectors are said to be coplanar
if their line segments lie in the same plane or are parallel to the same plane.
Show that the sum of the vectors represented by the
sides of a quadrilateral ABCD is equivalent
to the sum of the vectors represented by the diagonals and
From the triangular law of
- , addition of these two equation
the p.v. of a point P be and that of another
point Q be
the line joining P and Q is divided by a point R in the ratio of m:n
(internally or externally), then
For internal division take m : n as positive
and for external division take m : n as negative (i.e., either of m or n as
R is the mid-point of PQ, then
A, B, C are the vertices of a triangle with p.v.’s respectively,
then the p.v. of the centroid of the triangle is
given by .