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.
Representation of a vector:
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 |AB→| or |a→| (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° ≤ θ ≤ 180°