Normally, you use Cartesian coordinates
when dealing
with vectors. That is based on “Cartesian basis
vectors” . For example, in two
dimensions,
components, or
coefficients,” or “coordinatesof vector .
Therefore, in this way, you are never have to deal with more vectors that and . All the rest is just ordinary numbers.
But sometimes it is convenient to use a different basis than the
obvious one. For example, you might know that in
dealing with plane stresses, it is often convenient to rotate the
coordinate system to the principal axes.
In principal
axes, there are no shear stresses, just normal stresses. But if you
rotate the coordinate system, and become different
vectors, call them and . The point however is that
in using these new basis vectors and , your
physical problem has simplified.
As we will see later, to simplify a problem, the desired new vectors
are not always orthonormal (orthogonal and of length 1) like
and in the example above. In general, the new basis vectors,
we will call them and , can be anything, as long
as they are linearly independent. As long as that is true, you can
still write any arbitrary vector as
However, in general the coordinates and in the new basis are not the same as the coordinates and in the old basis . So, if you want to use the new basis to your advantage, you will normally have to know how to compute and if you know and and/or vice-versa. That is the problem that this section will address.
First, to find the old coordinates and given the new ones
and is easy. Just write the above equation as a
row-column multiplication:
(5) |
So you get the following transformation formulae between coordinates
(6) |
old to newreally means old from new.
The final thing you need to know is what happens to matrices. If a
matrix converts a vector to a vector in terms of
Cartesian coordinates, then should convert to
in terms of the new coordinates. Since
So the transformation rules for matrices are
(7) |
While we used a two dimensional example, you can do all of the above in any number of dimensions. You just add more basis vectors to transformation matrix .