Graphics Reference
In-Depth Information
Now let's see what
a
×
b
×
c
expands to. With reference to the above argument, we can state
that
i
j
k
a
×
b
×
c
=
y
a
z
b
−
y
b
z
a
z
a
x
b
−
z
b
x
a
x
a
y
b
−
x
b
y
a
x
c
y
c
z
c
Expanding gives
a
×
b
×
c
=
z
c
z
a
x
b
−
z
b
x
a
−
y
c
x
a
y
b
−
x
b
y
a
i
+
x
c
x
a
y
b
−
x
b
y
a
−
z
c
y
a
z
b
−
y
b
z
a
j
+
y
c
y
a
z
b
−
y
b
z
a
−
x
c
z
a
x
b
−
z
b
x
a
k
Expanding again gives
a
×
b
×
c
=
x
b
z
a
z
c
−
x
a
z
b
z
c
−
x
a
y
b
y
c
+
x
b
y
a
y
c
i
+
x
a
x
c
y
b
−
x
b
x
c
y
a
y
a
z
b
z
c
+
y
b
z
a
z
c
j
+
y
a
y
c
z
b
−
y
b
y
c
z
a
−
x
b
x
c
z
a
+
x
a
x
c
z
b
k
Again we split the above expansion as closely as possible into the form
a
+
b
by isolating the
a
and
b
components:
a
×
b
×
c
=
x
b
y
a
y
c
+
z
a
z
c
i
−
x
a
y
b
y
c
+
z
b
z
c
i
+
y
b
x
a
x
c
+
z
a
z
c
j
−
y
a
x
b
x
c
+
z
b
z
c
j
+
z
b
x
a
x
c
+
y
a
y
c
k
−
z
a
x
b
x
c
+
y
b
y
c
k
(B.5)
Once more, something's missing. Looking carefully at the expressions y
a
y
c
+
z
a
z
c
, x
a
x
c
+
z
a
z
c
,
and x
a
x
c
+
y
a
y
c
in Eq. (B.5), we notice that they are similar to the dot-product expansion,
except each has a missing term. These missing terms are x
a
x
c
, y
a
y
c
, and z
a
z
c
, respectively, and
can be introduced by adding
x
b
x
a
x
c
i
+
y
b
y
a
y
c
j
+
z
b
z
a
z
c
k
to the RHS of Eq. (B.5). Fortunately, x
b
x
a
x
c
i
z
b
z
a
z
c
k
has to be subtracted
from the RHS of Eq. (B.5) to convert the remaining terms into dot-product form. Thus, we get
+
y
b
y
a
y
c
j
+
×
×
=
x
b
x
a
x
c
+
y
a
y
c
+
+
y
b
x
a
x
c
+
y
a
y
c
+
+
z
b
x
a
x
c
+
y
a
y
c
+
a
b
c
z
a
z
c
i
z
a
z
b
j
z
a
z
b
k
−
x
a
x
b
x
c
+
y
b
y
c
+
−
y
a
x
b
x
c
+
y
b
y
c
+
−
z
a
x
b
x
c
+
y
b
y
c
+
z
b
z
c
i
z
b
z
c
j
z
b
z
c
k
which contains dot products:
a
×
b
×
c
=
x
b
a
·
c
i
+
y
b
a
·
c
j
+
z
b
a
·
c
k
−
x
a
b
·
c
i
−
y
a
b
·
c
j
−
z
a
b
·
c
k
(B.6)
