1 Vector algebra
TABLE OF CONTENTS
1. Introduction
2. Vector operations
2.1 Addition of vectors
2.2 Multiplication of a vector by a scalar
2.3 Dot product of two vectors
2.4 Cross product of two vectors
3. Component form of vectors
4. The triple products
5. Position and displacement vectors
5.1 Scalar and vector fields
LEARNING OBJECTIVES
1. An introduction to vectors with elementary definition and examples is given.
2. Various vector operations like addition, multiplication by a scalar, dot and cross products are defined and their properties discussed.
3. Vectors are written in terms of their component form in Cartesian coordinates.
4. Higher order vector products are introduced and some important relations are proved.
5. Two very special vectors, the position vector and the displacement vector, are introduced. Vector and scalar fields are defined.
V e c t o r A l g e b r a
1. Introduction
Students of physics and engineering are very familiar with the notion of vectors. However, for the sake of completeness we shall begin our study from beginning with the definition of vectors and their simple properties. Many physical quantities can be described adequately by their magnitude alone. Mass, temperature and potential are some of the quantities that fall in this category. Such quantities are called scalars. On the other hand there are quantities for whose description a magnitude is not enough; we need a direction as well. Displacement, velocity, force, electric and magnetic fields etc. are obvious examples. The result of a displacement towards the North is obviously different from that of a displacement toward the East. Similarly, the effect of a force acting on a body depends not only on its magnitude but also on the direction in which it acts.
Geometrically vectors are denoted by arrows; length of the arrow denoting its magnitude and the arrowhead its direction.
Note: Vectors have magnitude and direction but no location. Thus on a diagram if we slide a vector around; it remains the same vector as long as the magnitude and direction remain the same.
Equality of vectors:
Two vectors are said to be equal if and only if they have the same magnitude and direction.
2. Vector operations
We now define the basic vector operations – addition and multiplication of vectors. Since vectors have both magnitude and direction their laws of addition and multiplication are different from those of ordinary numbers. In particular we have three types of vector multiplications as we shall see now.
2.1 Addition of vectors
2.2 Multiplication of a vector by a scalar
Linear independence
Thus in a plane there are two linearly independent vectors. Similarly in space there are three linearly independent vectors. Any three non-coplanar vectors can be taken as the basis; then any vector can be written as a linear combination of these three. We usually choose three mutually perpendicular unit vectors as our basis vectors.
2.3 Dot product of two vectors
The dot product of two vectors is often also called the scalar product. In fact we shall use both the nomenclatures. The dot or scalar product of two vectors is a scalar quantity given by the product of the magnitudes of the two vectors and the cosine of the angle between them. If θ is the angle between the two vectors then
2.4 Cross product of two vectors
It is clear from the definition that the vector or cross product is not commutative. In fact it is anti-commutative:
3. Component form of vectors
So far we have dealt with vectors in an “abstract” form. However from a practical point of view it is much more convenient to deal with vectors in a “component” form. The vectors that we have introduced are vectors in our physical three dimensional space. (In physics there are many other vectors which need generalized vector spaces for their study. However that is another story; we are restricting ourselves to vectors in the physical space only.)
Thus we have the rule:
- To add vectors add like components.
Similarly follows the rule
- To multiply a vector by a scalar, multiply each component of the vector by that scalar.
On using properties (3) of the unit vectors we have
For the cross product in the component form we use properties (4) of the unit vectors and obtain
This looks rather cumbersome but can be put in a neater form by using determinants:
4. The triple products
Note that the cyclic order in which the three vectors appear must be preserved; otherwise there is a change of sign.
In component form the triple scalar product can be written as:
From equation (11) and the commutative property of scalar product, it follows that the dot and the cross in the product can be interchanged:
Triple vector product
Hence
- The triple vector product can be simplified to the form:
Here we prove this very important result by purely vector methods, though a much simpler proof is provided by the tensor method.
Using this in the triple vector product we have
Finally on solving these two equations for b and c, and substituting in equation (20) we get
All higher order vector products can be similarly simplified to terms containing only single vector products. For example
Reciprocal set of vectors
Proof
We give part of the proof here.
5. Position and displacement vectors
Two vectors of special importance in physics are the position vector and the related displacement vector. Most of the time we deal with vector fields, i.e. vectors that are functions of position and time. For example we have the potential produced by a point charge, or the temperature variation in a room. These are examples of scalar fields. Variation in electric and magnetic fields produced by a moving point charge, or the gravitational field due to a system of masses are examples of vector fields. A scalar or a vector field may in addition depend upon time. If it is independent of time we call it a static or stationary field.
SUMMARY
- In this module we have introduced the concept of vector quantities.
- Vectors are different from scalars and have different properties. We define the various vector operations like addition, multiplication by a scalar, dot and cross products and discuss their properties.
- We introduce a Cartesian coordinate system and see how vectors can be written in terms of their components, which is a more convenient way of working with them.
- We introduce triple vector products which appear often in the mathematical study of physics of materials and prove some important results.
- We next introduce two very special vectors, the position and displacement vectors. Their importance stems from the fact that more often we deal with vector and scalar fields, i.e., vectors and scalars that are functions of the position vectors.