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Directed line segment: A line segment with an arrowhead showing direction is called a directed line segment. Its two endpoints are distinguished as initial (or tail) and terminal (or head). The length of the line segment is called a magnitude of the directed line segment. See Fig.(i). The same line segment with arrowhead in the opposite direction is also a directed line segment. See Fig.(ii).

Vector: A quantity having both magnitude and direction is called a vector. Ex: displacement, velocity, acceleration, force, etc. Notice that a directed line segment is a vector (Fig.(iii)), denoted as or simply as and read as 'vector ' or 'vector '.

The point P from where the vector starts is called initial point and the point Q where it ends is called its terminal point. The distance between P and Q is called the magnitude of , denoted as ||, or ||, or 'p'. The arrow indicates the direction of the vector.

Position Vector

A vector having the origin of the chosen co-ordinate system (either 2D or 3D) as initial point and a point P as terminal point is called the position vector of P in that co-ordinate system. It is denoted by (or ).

If P(x, y) is a point in a plane, then the magnitude of is given by: || = Similarly, if P(x, y, z) is a point in a space, then the magnitude of is given by: || =

Types of Vectors

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.

Collinear vectors: Two or more vectors are said to be collinear if they are parallel to each other, irrespective of their magnitudes and directions.

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).

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