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which we recognise as the dot product
c
·
n
. But as
n
=
a
×
b
, we can state
=
·
×
Volume
c
a
b
which is a rather elegant relationship.
n
c
α
b
Area
θ
a
Figure 2.25.
The prism in Fig. 2.25 could easily be drawn with vectors
b
and
c
as the base, which, using the
above reasoning, would make
Volume
=
a
·
b
×
c
Similarly, if vectors
c
and
a
form the base, the volume is
Volume
=
b
·
c
×
a
thus, we can conclude that
Volume
=
a
·
b
×
c
=
b
·
c
×
a
=
c
·
a
×
b
(2.14)
The first two entries in the first column of Table 2.3 correspond to the volume calculating
qualities of this
scalar triple product
.
Unlike the vector product, the dot product is commutative, which permits us to write
c
·
a
×
b
=
a
×
b
·
c
which, using Eq. (2.14), leads to
a
·
b
×
c
=
a
×
b
·
c
(2.15)
and as Eq. (2.15) only makes sense when the parentheses embrace the cross product, we employ
abc
to represent
a
·
b
×
c
or
a
×
b
·
c
, which is also called the
box product
. However, the
