VECTOR
OPERATIONS
13
Now let’s repeat
this
derivation using the subscriptlsummation notation. Equa-
tion
l
.40
allows us to write
_-
A.
A
=
AiAi
(1.60)
(1.61)
A
.A
=
A:A:.
Notice how we have been careful to use different subscripts for the two sums in
Equations 1.60 and 1.61. This ensures the
sums
will remain independent as they
are manipulated in the following steps. The primed components can be expressed in
terms
of
the unprimed components as
-1 -1
A,!
=
R..A
‘I
I
.,
(1.62)
where
Rij
is the
ijth
component of the rotation matrix
R[4].
Inserting this expression
into Equation 1.61 gives
(1.63)
A
*A
=
R,A,R,,A,,
where again, we have been careful to use
the
two different subscripts
u
and
v.
This
equation has three implicit sums, over the subscripts
r,
u,
and
u.
In
subscript notation, unlike matrix notation, the ordering
of
the terms is not
important,
so
we rearrange Equation 1.63 to read
--I --I
(1.64)
A *A
=
A,A,R,R,,.
Next concentrate on the sum over
r,
which only involves the
[R]
matrix elements,
in the product
R,R,,.
What exactly does
this
product mean? Let’s compare it to an
operation we discussed earlier. In Equation 1.12, we pointed out the subscripted ex-
pression
MijNjk
represented the regular matrix product
[M][N],
because the summed
subscript
j
is in the second position
of
the
[MI
matrix and the first position of the
[N]
matrix. The expression
R,R,,,
however, has a contraction over the first index
of
both
matrices. In order to make sense
of
this product, we write the first instance
of
[R]
using the transpose:
--I
--I
RruRru
+
[Rlt [Rl-
(1.65)
Consequently, from Equation 1.57,
R,R,,
=
&,.
(1.66)
Substituting this
result
into Equation 1.64 gives
(1.67)
Admittedly, this example is too easy. It
does
not demonstrate any significant
advantage of using the subscriptlsummation notation over matrices. It does, how-
ever, highlight the equivalence
of
the two approaches. In our next example, the
subscriptlsummation notation will prove to
be
almost indispensable.
