Graphics Reference
In-Depth Information
Substituting
b
+
c
for
d
,weget
y
a
z
a
z
a
x
a
x
a
y
a
a
×
b
+
c
=
i
+
j
+
k
y
b
+
y
c
z
b
+
z
c
z
b
+
z
c
x
b
+
x
c
x
b
+
x
c
y
b
+
y
c
a
×
b
+
c
=
y
a
z
b
+
z
c
−
z
a
y
b
+
y
c
i
+
z
a
x
b
+
x
c
−
x
a
z
b
+
z
c
j
+
x
a
y
b
+
y
c
−
y
a
x
b
+
x
c
k
a
×
b
+
c
=
y
a
z
b
−
y
b
z
a
i
+
y
a
z
c
−
y
c
z
a
i
+
z
a
x
b
−
z
b
x
a
j
+
z
a
x
c
−
z
c
x
a
j
+
x
a
y
b
−
x
b
y
a
k
+
x
a
y
c
−
x
c
y
a
k
a
×
b
+
c
=
a
×
b
+
a
×
c
2.10 Triple products
Having examined the products that involve two vectors, now let's look at the products that
involve three vectors. An interesting approach is to take three vectors,
a
,
b
,
c
, and explore the
combinations using a '
' without disturbing the alphabetic order of the vector names.
These are summarised in Table 2.3.
·
'anda'
×
Table 2.3
a
·
×
×
·
b
c
a
b
c
a
×
b
·
c
a
·
b
×
c
a
×
b
×
c
a
·
b
·
c
a
×
b
×
c
a
·
b
·
c
Not all the combinations in Table 2.3 lead to a meaningful result, and for neatness the
meaningless combinations are arranged in the second column. They are meaningless because
the operation in parentheses is a dot product, which is a scalar, and cannot be associated with
a scalar product or vector product. So we are left with the four triple products in the first
column, which do satisfy the rules associated with the scalar and vector products. But what do
they mean? Well, let's find out.
2.10.1 Scalar triple product
Figure 2.25 shows three vectors,
a
,
b
, and
c
, that provide the basis for a prism, where the area of
the base is given by Area
=
a
b
sin. But
n
=
a
×
b
, where
n
=
Area. Now the volume
of a prism is the product of its (vertical height)
·
(base area), which in vector notation is given by
Volume
=
c
cos
·
n
=
c
n
cos
