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Coordinate Transformations
A Cartesian coordinate system allows position and direction in space to be represented in a very convenient
manner. Unfortunately, such a coordinate system also introduces arbitrary elements into our analysis. After all, two independent observers might well choose coordinate systems with different origins, and
different orientations of the coordinate axes. In general, a given vector will have different
sets of components in these two coordinate systems. However, the direction and magnitude of are the
same in both cases. Hence, the two sets of components must be related to one another in a very particular fashion.
Actually, since vectors are represented by moveable line elements in space (i.e., in Figure A.97,
and
represent the same vector), it follows that
the components of a general vector are not affected by a simple shift in the origin of a Cartesian coordinate system. On the other hand, the
components are modified when the coordinate axes are rotated.
Suppose that we transform to a new coordinate system, , which has the
same origin as , and is obtained by
rotating the coordinate axes of through an angle
about see Figure A.100.
Let the coordinates of a general point be in and
in . According to simple trigonometry, these two sets of coordinates are related to one another via the transformation:
Consider the vector displacement
.
Note that this displacement is represented by the same symbol, , in both
coordinate systems, since the magnitude and
direction of are manifestly independent of the orientation of the coordinate axes. The
coordinates of do depend on the orientation of the axes: i.e.,
in , and
in .
However, they must depend in a very specific manner [i.e., Equations (A.1277)(A.1279)] which
preserves the magnitude and direction of .
Figure A.100:
Rotation of the coordinate axes about .

The
components of a general vector transform in an analogous
manner to Equations (A.1277)(A.1279): i.e.,
Moreover, there are similar transformation rules for rotation about and .
Equations (A.1280)(A.1282) effectively constitute the definition of a vector: i.e.,
the three
quantities (
) are the components of a vector provided that
they transform under rotation of the coordinate axes about in accordance with Equations (A.1280)(A.1282). (And also transform
correctly under rotation about and ).
Conversely, (
) cannot be the components of a vector if they
do not transform in accordance with Equations (A.1280)(A.1282). Of course, scalar quantities are invariant
under rotation of the coordinate axes.
Thus, the individual components of a vector (, say) are real numbers, but
they are
not scalars.
Displacement vectors, and all vectors derived from
displacements (e.g., velocity, acceleration), automatically satisfy Equations (A.1280)(A.1282). There are, however, other
physical quantities which have both magnitude and direction, but which are not
obviously related to displacements. We need to check carefully to see whether these
quantities are really vectors (see Section A.8).
Next: Scalar Product
Up: Vector Algebra and Vector
Previous: Cartesian Components of a
Richard Fitzpatrick
20110331