This Quantum World/Appendix/Vectors

Vectors (spatial)Edit

A vector is a quantity that has both a magnitude and a direction. Vectors can be visualized as arrows. The following figure shows what we mean by the components   of a vector  

 

The sum   of two vectors has the components  

  • Explain the addition of vectors in terms of arrows.

The dot product of two vectors is the number

 

Its importance arises from the fact that it is invariant under rotations. To see this, we calculate

 
 

According to Pythagoras, the magnitude of   is   If we use a different coordinate system, the components of   will be different:   But if the new system of axes differs only by a rotation and/or translation of the axes, the magnitude of   will remain the same:

 

The squared magnitudes     and   are invariant under rotations, and so, therefore, is the product  

  • Show that the dot product is also invariant under translations.

Since by a scalar we mean a number that is invariant under certain transformations (in this case rotations and/or translations of the coordinate axes), the dot product is also known as (a) scalar product. Let us prove that

 

where   is the angle between   and   To do so, we pick a coordinate system   in which   In this coordinate system   with   Since   is a scalar, and since scalars are invariant under rotations and translations, the result   (which makes no reference to any particular frame) holds in all frames that are rotated and/or translated relative to  

We now introduce the unit vectors   whose directions are defined by the coordinate axes. They are said to form an orthonormal basis. Ortho because they are mutually orthogonal:

 

Normal because they are unit vectors:

 

And basis because every vector   can be written as a linear combination of these three vectors — that is, a sum in which each basis vector appears once, multiplied by the corresponding component of   (which may be 0):

 

It is readily seen that       which is why we have that

 

Another definition that is useful (albeit only in a 3-dimensional space) is the cross product of two vectors:

 
  • Show that the cross product is antisymmetric:  

As a consequence,  

  • Show that  

Thus   is perpendicular to both   and  

  • Show that the magnitude of   equals   where   is the angle between   and   Hint: use a coordinate system in which   and  

Since   is also the area   of the parallelogram   spanned by   and   we can think of   as a vector of magnitude   perpendicular to   Since the cross product yields a vector, it is also known as vector product.

(We save ourselves the trouble of showing that the cross product is invariant under translations and rotations of the coordinate axes, as is required of a vector. Let us however note in passing that if   and   are polar vectors, then   is an axial vector. Under a reflection (for instance, the inversion of a coordinate axis) an ordinary (or polar) vector is invariant, whereas an axial vector changes its sign.)

Here is a useful relation involving both scalar and vector products: