An example of a vector space that is not an Inner Product Space

An example of a vector
in which it is not useful to define a length, consider the example of geometrical
optics. It is useful (approximations
of Gaussian optics) to define a vector in which one component is the distance, *h *,
away from the optical axis (a line running through the center of the optical
system) and a second component is the angle, *θ *, the ray makes
with the axis. We can add two of these vectors, or multiply them by scalars,
and we have inverses as required by the definition of a linear
space.

The value of this representation appears when we consider
how we can represent the propagation
of the ray or to bending by a lens. In the small angle approximation (where
sin *θ* ≈ *θ*), these actions
can be described in terms of this two component vector ( *h*, *θ*)
as the action of matrices as seen in the figures below.

Propagation of a ray by
a distance *d *(in the small angle approximation): the angle stays the
same, the height changes.

Refraction of a ray at
a thin lens of focal length *f*: the height stays the same, the angle changes.

The "length" of this vector is meaningless since you cannot add a length and an angle.

University of Maryland | Physics Department | Physics 374 Home |
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This page prepared by Edward F. Redish

Department of Physics

University of Maryland

College Park, MD 20742

Phone: (301) 405-6120

Email: redish@umd.edu

Last revision 27. October, 2005.