Introduction to vectors

Introduction to vectors: In mathematics we have many systems which are employed to handle problems in Geometry, Mechanics and other branches of Applied Mathematics. Vectors constitute one of these systems. Vectors facilitate analytic study of the previous type of physical objects which have direction in addition to magnitude. No doubt, the set of real numbers provides an analytical tool for study of various types of physical problems for which vectors are useful.

The use of vectors in these problems is more natural. We have to associate a physical entity involving direction to the set of real numbers for we have to split up the entity into components and associate a number with each. The use of vectors avoids this splitting up. What is a vector? Quantities that have magnitude as well as direction are called vectors.

Definition of a vector: A directed line segment is called a vector. A vector from P to Q is denoted by  . P and Q are called respectively initial and terminal points of the vector  . Vector Example: Such quantities are called vectors Displacement, velocity, acceleration, momentum, weight, force etc.

Types of vectors are Zero or null vector: A vector whose initial and terminal points are coincident is called the zero or the null vector. Vectors other than the null vector are called proper vectors.

Unit vector: A vector whose modulus is unity is called the unit vector.

Like and unlike vectors: Vectors are said to be like when they have the same sense of direction and unlike when they have opposite directions.

Collinear or parallel vectors: Vectors having the same or parallel supports are called collinear vectors.

Co-initial vectors: Vectors having the same initial point are called co-initial vectors.

Scalars quantities: Quantities that have only magnitude but no direction are called scalars. For example, time, mass, volume, population, temperature, energy, power are all scalars. We need a unit and a real number to specify a scalar. For example, 3 hours,  2.5 kg, 8 cubic centimetres, 23°F etc.

Vector and scalar quantities

Physical quantities are divided into two categories- Vector and scalar. Those quantities which have only magnitude and which are not related to any fixed direction in space are called scalar quantities or scalars. Example of scalars are mass, volume, density, work, temperature etc. Second kinds of quantities are those which have both magnitude and direction.


Co-planar vectors: A system of vectors is said to be coplanar, if their supports are parallel to the same plane. Two vectors are always coplanar.