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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 righthanded if, when looking along the direction, a clockwise
rotation about is required to take into . Otherwise, it is said to be lefthanded. In physics, it is conventional to always use righthanded coordinate systems.
Figure A.99:
A righthanded Cartesian coordinate system.

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

(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

(1272) 
where , , and are termed the Cartesian components of . It is coventional to write
.
It follows that
,
, and
. Of course,
.
According to the threedimensional generalization of the Pythagorean theorem, the distance
is
given by

(1273) 
By analogy, the magnitude of a general vector takes the form

(1274) 
If
and
then it is
easily demonstrated that

(1275) 
Furthermore, if is a scalar then it is apparent that

(1276) 
Next: Coordinate Transformations
Up: Vector Algebra and Vector
Previous: Vector Algebra
Richard Fitzpatrick
20110331