A 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.

Pictorially, we
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.** **

The 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.

**Note:**

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.

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.

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 £ 180°

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= , (-1) = -, etc.

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 k_{1} (k_{2}) = (k_{1}k_{2}) = k_{2}(k_{1}) " k_{1}, k_{2 }∈ 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).

Given 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, = ; + = ;

(k_{1} + k_{2})= k_{1}+ k_{2}; k (=
k+ k.

**Remarks: **

- Two
vectors (non-zero and non-collinear) constitute a 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

and

Since =
- , addition of these two equation
gives, .** **

Let the p.v. of a point P be and that of another point Q be

If 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 negative). .

If R is the mid-point of PQ, then

If
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 .