1.1 The Vector Concept Revisited

The concept of a vector has been one of the most fruitful ideas in all of mathematics, and it is not surprising that we receive repeated exposure to the idea throughout our education. Students in elementary mathematics deal with vectors in component form with quantities such as

for example. But let us examine this situation more closely. Do the components 2, 1, 3 determine the vector x? They surely do if we specify the basis vectors of the coordinate frame. In elementary mathematics these are supposed to be mutually orthogonal and of unit length; even then they are not fully characterized, however, because such a frame can be rotated. In the description of many common phenomena we deal with vectorial quantities like forces that have definite directions and magnitudes. An example is the force your body exerts on a chair as you sit in front of the television set. This force does not depend on the coordinate frame employed by someone writing a textbook on vectors somewhere in Russia or China. Because the vector f representing a particular force is something objective, we should be able to write it in such a form that it ceases to depend on the details of the coordinate frame. The simplest way is to incorporate the frame vectors e _i, i=1, 2, 3, explicitly into the notation: if x is a vector we may write

(1.1)

Then if we wish to change the frame, we...