Zero vector: A vector whose initial and terminal points coincide, is called a zero vector or null vector. In other words, a vector that has zero magnitude is called a zero vector. It is denoted by or , , etc. The direction of the zero vector is indeterminate.
Unit vector: A vector whose magnitude is 1 unit (i.e., unity) is called a unit vector. The unit vector in the direction of a given vector is represented by and read as 'b cap' or 'b hat'.
The only purpose of unit vectors is to describe the direction. In Cartesian coordinate system, , and are the unit vectors that point along the x, y and z-axes respectively.
Co-initial vectors: The vectors having the same initial point are called co-initial vectors.
Equal vectors: Two vectors and are said to be equal, if they have the same magnitude (i.e., || = ||) and direction regardless of whether they have the same initial points or not. If and are equal vectors, then = .
Negative of a vector: A vector having same magnitude but in the opposite direction to a given vector is called the negative vector. For example, vector is negative of the vector and written as: = – . In order to find the negative of a vector, we merely reverse its arrowhead (direction).
If a1, a2, a3 are three real numbers, then the ordered triad
(a1, a2, a3) is called a space vector.
The real numbers a1, a2, a3
are called first, second and third components of the space vector (a1, a2, a3).
Ex: The 3D co-ordinates of an object in space: x, y and z
A vector which is independent of its position is called a free vector.
Ex: The direction of motion of a wave.
If a is a vector and P is a point, then the ordered pair (P, a) is called localised vector at P.
Ex: The position vector is a localised vector or a fixed vector.
If a is a vector and P is a point, then there exists a point Q such that a = PQ.
The vector PQ is called localized vector of a at P.
Like and unlike vectors:
Two vectors are said to be like vectors if they have the same direction.
Ex: An express train overtaking a passenger train
Two vectors are said to be unlike vectors if they are in opposite directions.
Ex: Positive and negative co-ordinate axes