San José State University

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Thayer Watkins
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The Variety of Vector and Tensor Types

## What is a vector?

A three dimensional vector is represented in a particular coordinate system by a triplet of real numbers. But the vector is not that triplet of numbers, it is something whose representation as three numbers changes in a systematic way as the coordinate system changes.

What is ordinarily meant by the term vector is called a polar vector. There is also something called an axial vector, which is the vector (cross) product of two polar vectors. The difference between polar and axial vectors is revealed when we consider a transformation of the coordinate system that changes the handedness of the coordinate system.

Consider two vectors whose respresentation are:

A = (1, 2, 3) and B = (3, 4, 7).

Their cross product C=A×B is: (2, 2, -4).

If we invert the coordinate system; i.e.,

#### x → -x,  y → -y,  z → -z,

then the representation of A becomes
(-1, -2, -3) and likewise the representation of B becomes (-3, -4, -7). (If the original coordinate system was right-handed then the new coordinate system is left-handed.) If C were a true vector then its representation should go to (-2. -2, 4). But the cross product of the representations of A and B in the new coordinate system is (2, 2, -4). Thus representation of C = AxB did not go to (-2, -2, 4) as would be required if C were to be a true vector. For this reason C=AxB is called a pseudovector or axial vector.

An axial vector can also be considered a representation of a second order antisymmetric tensor; i.e.,

 B1 B2 B3
 →
 0 B1 B2 −B1 0 B3 −B2 −B3 0

The variety of vector and tensor concepts are shown below. There is even some variety of the scalar concept.

Types of Scalars, Vectors and Tensors
NameOriginExamplesTransformation Law
Polar VectorOrdinary VectorDisplacement,
Electric Field
E'i = ΣlirEr
Axial VectorCross product
of polar vectors
Magnetic Field B'i=
|det (l)|Σlir Br
Polar TensorRelation between
two vectors
of the same type
Magnetic Permeability P'i..j=
Σlir ...lkwPr..w
Axial TensorRelation between
two vectors of
different types
Optical Gyration Q'i..j=|det(l)|Σlir ...lkwQr..w
Psuedo-scalarpolar vector
dotted with
axial vector
Rotary Power of
Optically Active
Crystal

(To be continued.)