Biomedical Engineering Reference
In-Depth Information
a
b
¼
e
jik
a
j
b
i
e
k
;
Next change
e
jik
to
e
ijk
and rearrange the order of
a
j
and
b
i
, then the result is
proved:
a
b
¼
e
ijk
b
i
a
j
e
k
¼
b
a
:
Æ
Scalar Triple Product of Three Vectors
The scalar triple product of three vectors is a scalar formed from three vectors,
a
ð
b
c
Þ
and the triple vector product is a vector formed from three vectors, (
r
(
p
q
)). An expression for the scalar triple product is obtained by taking the dot
product of the vector
c
with the cross product in the representation (A.114) for
a b
,thus
c
ð
a
b
Þ¼
e
jik
a
j
b
i
c
k
:
(A.116)
From the properties of the alternator it follows that
c
ð
a
b
Þ¼
a
ð
b
c
Þ¼
b
ð
c
a
Þ¼
a
ð
c
b
Þ¼
b
ð
a
c
Þ
¼
c
ð
b
a
Þ:
(A.117)
If the three vectors
a
,
b
, and
c
coincide with the three nonparallel edges of
a parallelepiped, the scalar triple product
a ðb cÞ
is equal to the volume of the
parallelepiped. In the following example a useful vector identity for the triple vector
product (
r
(
p
q
)) is derived.
Example A.8.4
Prove that (
r
(
p
q
))
¼
(
r
q
)
p
(
r
p
)
q
.
Solution: First rewrite (A.114) with the change
a
!
r
, and again with the changes
a
!
p
and
b
!
q
, where
b
¼
(
p
q
)
r
b
¼
e
ijk
r
i
b
j
e
k
;
b
¼
p
q
¼
e
mnj
p
m
q
n
e
j
;
Note that the second of these formulas gives the components of
b
as
b
j
¼
e
mnj
p
m
q
n
:
This formula for the components of
b
is then substituted into the expression for
(
r
b
)
¼
(
r
(
p
q
)) above, thus
r
ð
p
q
Þ¼
e
ijk
e
mnj
r
i
p
m
q
n
e
k
:
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