1.4 Vectors  (Page 2/5)

Mathematical concept of vector is basically secular in nature and general in application. This means that mathematical treatment of vectors is without reference to any specific physical quantity or phenomena. In other words, we can employ vector and its methods to all quantities, which possess directional attribute, in a uniform and consistent manner. For example two vectors would be added in accordance with vector addition rule irrespective of whether vectors involved represent displacement, force, torque or some other vector quantities.

The moot point of discussion here is that vector has been devised to suit the requirement of natural process and not the other way around that natural process suits vector construct as defined in vector mathematics.

What is a vector?

Vector

Vector is a physical quantity, which has both magnitude and direction.

A vector is represented graphically by an arrow drawn on a scale as shown Figure i . In order to process vectors using graphical methods, we need to draw all vectors on the same scale. The arrow head point in the direction of the vector.

A vector is notionally represented in a characteristic style. It is denoted as bold face type like “ $\mathbf{a}$ ” as shown Figure (i) or with a small arrow over the symbol like “ $\overrightarrow{a}$ ” or with a small bar as in “ $\stackrel{-}{a}$ ”. The magnitude of a vector quantity is referred by simple identifier like “a” or as the absolute value of the vector as “ $\left|\mathbf{a}\right|$ ” .

Two vectors of equal magnitude and direction are equal vectors ( Figure (ii) ). As such, a vector can be laterally shifted as long as its direction remains same ( Figure (ii) ). Also, vectors can be shifted along its line of application represented by dotted line ( Figure (iii) ). The flexibility by virtue of shifting vector allows a great deal of ease in determining vector’s interaction with other scalar or vector quantities.

It should be noted that graphical representation of vector is independent of the origin or axes of coordinate system except for few vectors like position vector (called localized vector), which is tied to the origin or a reference point by definition. With the exception of localized vector, a change in origin or orientation of axes or both does not affect vectors and vector operations like addition or multiplication (see figure below).

The vector is not affected, when the coordinate is rotated or displaced as shown in the figure above. Both the orientation and positioning of origin i.e reference point do not alter the vector representation. It remains what it is. This feature of vector operation is an added value as the study of physics in terms of vectors is simplified, being independent of the choice of coordinate system in a given reference.

Vector algebra

Graphical method is slightly meticulous and error prone as it involves drawing of vectors on scale and measurement of angles. In addition, it does not allow algebraic manipulation that otherwise would give a simple solution as in the case of scalar algebra. We can, however, extend algebraic techniques to vectors, provided vectors are represented on a rectangular coordinate system. The representation of a vector on a coordinate system uses the concept of unit vectors and scalar magnitudes. We shall discuss these aspects in a separate module titled Components of a vector . Here, we briefly describe the concept of unit vector and technique to represent a vector in a particular direction.