Introduction to Classical Mechanics (Paperback)

Vector Algebra

In physics, we try to find relationships between various physical quantities whose values may be determined experimentally.
Many quantities in physics can be completely specified by giving their magnitude alone. Such quantities are mass, density, temperature, etc. These are called scalar quantities. Many other quantities however, require, in addition to their magnitude, direction for their complete speci- fication. These are called vector quantities. Displacement, force, electric field intensity, etc., are examples of vector quantities.
Many equations in physics assume a compact form when written in vector notation. The relationship between various quantities involved in equa- tion is revealed immediately when these are written in vector form. Histori- cally, vector notation became widely used with the advent of Maxwell's electromagnetic theory in which the above advantages are clearly seen.
Sometimes, we come across physical phenomenon in which we can asso- ciate a particular value of the variable with each point in a given region of the space. Such a region of the space is called a field. If the variable describes a scalar quantity, the field is called a scalar field. For example a temperature field around a hot body. If the variable describes a vector quantity, the field is called a vector field, for example a magnetic field.
A vector quantity may be geometrically represented by a straight line (i) having a length proportional to the magnitude of the vector quantity, and (ii) drawn in the same direction and sense as that of the given vector quantity.
In this book, we shall use the following notation for the representation of the vector quantities: (i) Bold-faced letters are used to represent the vector quantities. Thus, A represents 'vector A'. Similarly, PQ means a vector represented by a segment PQ of a straight line directed from P to 2. (ii) While writing the magnitude of vector quantities, italic letters are used. Thus, A represents the magnitude of vector A.