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Consider a Cartesian coordinate system
consisting of
an origin,
, and three mutually perpendicular coordinate axes,
,
, and
--see Figure A.99. Such a system is said to be right-handed if, when looking along the
direction, a
clockwise
rotation about
is required to take
into
. Otherwise, it is said to be left-handed. In physics, it is conventional to always use right-handed coordinate systems.
Figure A.99:
A right-handed Cartesian coordinate system.
![\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{AppendixA/figA.02b.eps}}
\end{figure}](img3216.png) |
It is convenient to define unit vectors,
,
, and
, parallel to
,
, and
, respectively.
Incidentally, a unit vector is a vector whose magnitude is unity. The position vector,
, of some general point
whose Cartesian coordinates
are (
,
,
) is then given by
![\begin{displaymath}
{\bf r} = x\,{\bf e}_z + y\,{\bf e}_y+z\,{\bf e}_z.
\end{displaymath}](img3217.png) |
(1271) |
In other words, we can get from
to
by moving a distance
parallel to
, then a distance
parallel to
, and then a distance
parallel to
. Similarly, if
is an arbitrary vector then
![\begin{displaymath}
{\bf a} = a_x\,{\bf e}_x+ a_y\,{\bf e}_y+a_z\,{\bf e}_z,
\end{displaymath}](img3218.png) |
(1272) |
where
,
, and
are termed the Cartesian components of
. It is coventional to write
.
It follows that
,
, and
. Of course,
.
According to the three-dimensional generalization of the Pythagorean theorem, the distance
is
given by
![\begin{displaymath}
r = \sqrt{x^2 + y^2 + z^2}.
\end{displaymath}](img3228.png) |
(1273) |
By analogy, the magnitude of a general vector
takes the form
![\begin{displaymath}
a = \sqrt{a_x^{\,2} + a_y^{\,2} + a_z^{\,2}}.
\end{displaymath}](img3229.png) |
(1274) |
If
and
then it is
easily demonstrated that
![\begin{displaymath}
{\bf a} + {\bf b} \equiv (a_x+b_x,\,a_y+b_y,\,a_z+b_z).
\end{displaymath}](img3231.png) |
(1275) |
Furthermore, if
is a scalar then it is apparent that
![\begin{displaymath}
n\,{\bf a} \equiv (n\,a_x,\,n\,a_y,\,n\,a_z).
\end{displaymath}](img3232.png) |
(1276) |
Next: Coordinate Transformations
Up: Vector Algebra and Vector
Previous: Vector Algebra
Richard Fitzpatrick
2011-03-31