2.13. The Scalar Triple Product
The scalar triple product is given by
A⋅(
B×C
)=
ε
ijk
a
i
b
j
c
k
=
a
1
a
2
a
3
b
1
b
2
b
3
c
1
c
2
c
3

= volume of parallelepiped with sides A, B and C.
Theorem 2.14
3 vectors A, B and C in R^{3}
are linearly dependent Û
A⋅(
B×C
)=0
.
Proof
Þ
If B
and C are linearly dependent, then
B×C=0
and theorem is proved.
If B
and C are L.I., then there exists
α,β,γ
,
not all zero, such that
αA+βB+γC=0
Furthermore
α≠0
since otherwise
α=β=γ=0
. Taking
(
B×C
)⋅
on both sides gives
αA⋅(
B×C
)=0
. QED.
Ü
Given
A⋅(
B×C
)=0
,
we have either
B×C=0
or A
is orthogonal to
B×C≠0
. For the former case, B and C are dependent and
the theorem is proved. For the latter
case, theorem 2.13 says that B, C and
B×C
form a basis so that
A=αB+βC+γB×C
Taking
(
B×C
)⋅
on both sides gives
0=γ
‖
B×C
‖
2
so that
γ=0
,
i.e.,
A=αB+βC
QED.