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Why should we study vectors?

The basic concepts in physics – particularly the branch of mechanics - have a direct and inherently characterizing relationship with the concept of vector. The reason lies in the directional attribute of quantities, which is used to describe dynamical aspect of natural phenomena. Many of the physical terms and concepts are simply vectors like position vector, displacement vector etc. They are as a matter of fact defined directly in terms of vector like “it is a vector ……………”.

The basic concept of “cause and effect” in mechanics (comprising of kinematics and dynamics), is predominantly based on the interpretation of direction in addition to magnitude. Thus, there is no way that we could accurately express these quantities and their relationship without vectors. There is, however, a general tendency (particular in the treatment designed for junior classes) to try to evade vectors and look around ways to deal with these inherently vector based concepts without using vectors! As expected this approach is a poor reflection of the natural process, where basic concepts are simply ingrained with the requirement of handling direction along with magnitude.

It is, therefore, imperative that we switch over from work around approach to vector approach to study physics as quickly as possible. Many a times, this scalar “work around” inculcates incorrect perception and understanding that may persist for long, unless corrected with an appropriate vector description.

The best approach, therefore, is to study vector in the backdrop of physical phenomena and use it with clarity and advantage in studying nature. For this reasons, our treatment of “vector physics” – so to say - in this course will strive to correlate vectors with appropriate physical quantities and concepts.

The most fundamental reason to study nature in terms of vectors, wherever direction is involved, is that vector representation is concise, explicit and accurate.

To score this point, let us consider an example of the magnetic force experienced by a charge, q, moving with a velocity v in a magnetic field, “ B . The magnetic force, F , experienced by moving particle, is perpendicular to the plane, P, formed by the the velocity and the magnetic field vectors as shown in the figure .

Magnetic force as cross product of vectors

The force is given in the vector form as :

F = q ( v x B )

This equation does not only define the magnetic force but also outlines the intricacies about the roles of the each of the constituent vectors. As per vector rule, we can infer from the vector equation that :

  • The magnetic force ( F ) is perpendicular to the plane defined by vectors v and B .
  • The direction of magnetic force i.e. which side of plane.
  • The magnitude of magnetic force is "qvB sinθ", where θ is the smaller angle enclosed between the vectors v and B .

This example illustrates the compactness of vector form and completeness of the information it conveys. On the other hand, the equivalent scalar strategy to describe this phenomenon would involve establishing an empirical frame work like Fleming’s left hand rule to determine direction. It would be required to visualize vectors along three mutually perpendicular directions represented by three fingers in a particular order and then apply Fleming rule to find the direction of the force. The magnitude of the product, on the other hand, would be given by qvB sinθ as before.

The difference in two approaches is quite remarkable. The vector method provides a paragraph of information about the physical process, whereas a paragraph is to be followed to apply scalar method ! Further, the vector rules are uniform and consistent across vector operations, ensuring correctness of the description of physical process. On the other hand, there are different set of rules like Fleming left and Fleming right rules for two different physical processes.

The last word is that we must master the vectors rather than avoid them - particularly when the fundamentals of vectors to be studied are limited in extent.

Questions & Answers

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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