Environmental Engineering Reference
In-Depth Information
Thus the distance is indeed the same in both frames and we can write Eq. (11.11) as
d
2
x
1
j
d
t
2
R
ji
(x
2
j
−
x
1
j
)
=
m
1
R
ji
Gm
1
m
2
x
1
k
)
]
3
/
2
.
[
(x
2
k
−
x
1
k
)(x
2
k
−
We are almost done, all that remains is to multiply either side by R
li
with the
implied summation over i whereupon we can use the fact that R
li
R
ji
=
δ
jl
,i.e.
d
2
x
1
l
d
t
2
(x
2
l
−
x
1
l
)
m
1
=
Gm
1
m
2
x
1
k
)
]
3
/
2
.
[
(x
2
k
−
x
1
k
)(x
2
k
−
And we have proven that the equation does not change its form under a rotation
of the co-ordinate system. We chose to perform this calculation explicitly in
component notation and hopefully you managed to thread your way through the
maze of indices. We could have worked in terms of matrices and column vectors,
in which case we write
d
2
(
R
−
1
r
)
T
(
R
−
1
r
)
=
r
T
(
R
−
1
)
T
(
R
−
1
)
r
=
r
T
RR
T
r
=
r
T
r
=
d
2
=
and Eq. (11.11) becomes
m
1
R
−
1
d
2
x
1
d
t
2
R
−
1
(
x
2
−
x
1
)
x
2
−
x
1
)
]
3
/
2
,
which reduces to the required form after multiplying both sides by
R
.
=
Gm
1
m
2
x
2
−
x
1
)
T
(
[
(
The previous example illustrates the usefulness of the scalar product between
two vectors since the proof of Eq. (11.12) can easily be broadened to show that
a
i
b
i
=
a
i
b
i
(11.13)
for any two vectors
. It should be stressed that this is not inevitable. There
are an infinity of ways in which two vectors can be combined to give a pure number
but only one way yields a pure number that is also scalar. For example, given our
two vectors we might combine them as
a
1
b
1
a
and
b
−
+
a
3
b
3
. The resultant number
is not a scalar quantity for its value does depend upon whether we are in frame
T
or
T
. So the scalar product is the only possible way to combine two vectors
in order to produce a scalar. There is likewise only one way to produce a vector
quantity from two vectors and that is the vector product. Specifically this means
that under the rotation
R
the vector product must necessarily satisfy
a
2
b
2
(
R
V
)
×
(
R
W
)
=
R
(
V
×
W
),
(11.14)
where
V
and
W
are any two vectors.
