Get the Knowledge that sets you free...Science and Math for K8 to K12 students

Login / Register

Login to your account

Email
Password
×

Warning

Please Login to Read More...

Introduction

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).

Flash is Not Installed in Your System. Please Click here to Install. Close
Java is Not Installed in Your System. Please Click here to Install. Close