Graphics Reference
In-Depth Information
Expanding gives
a
×
b
×
c
=
y
a
x
b
y
c
−
x
c
y
b
−
z
a
z
b
x
c
−
z
c
x
b
i
+
z
a
y
b
z
c
−
y
c
z
b
−
x
a
x
b
y
c
−
x
c
y
b
j
+
x
a
z
b
x
c
−
z
c
x
b
−
y
a
y
b
z
c
−
y
c
z
b
k
Expanding again gives
×
×
=
x
b
y
a
y
c
−
x
c
y
a
y
b
−
x
c
z
a
z
b
+
a
b
c
x
b
z
a
z
c
i
+
y
b
z
a
z
c
−
y
c
z
a
z
b
−
x
a
x
b
y
c
+
x
a
x
c
y
b
j
+
x
a
x
c
z
b
−
x
a
x
b
z
c
−
y
a
y
b
z
c
+
y
a
y
c
z
b
k
Now we split the above equation as closely as possible into the form
b
+
c
by isolating the
b
and
c
components:
a
×
b
×
c
=
x
b
y
a
y
c
+
z
a
z
c
i
−
x
c
y
a
y
b
+
z
a
z
b
i
+
y
b
z
a
z
c
+
x
a
x
c
j
−
y
c
x
a
x
b
+
z
a
z
b
j
+
z
b
x
a
x
c
+
y
a
y
c
k
−
z
c
x
a
x
b
+
y
a
y
b
k
(B.2)
We are almost there, but something's still missing. Looking carefully at the expressions
y
a
y
c
+
y
a
y
c
in Eq. (B.2), we notice that they are similar to the
dot-product expansion, except that each is missing a 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
z
a
z
c
, z
a
z
c
+
x
a
x
c
, and x
a
x
c
+
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.2). 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.2) to convert the remaining terms into dot-product form. Thus, we get
+
y
b
y
a
y
c
j
+
a
×
b
×
c
=
x
b
x
a
x
c
+
y
a
y
c
+
z
a
z
c
i
+
y
b
x
a
x
c
+
y
a
y
c
+
z
a
z
c
j
+
z
b
x
a
x
c
+
y
a
y
c
+
z
a
z
b
k
−
x
c
x
a
x
b
+
y
a
y
b
+
z
a
z
b
i
−
y
c
x
a
x
b
+
y
a
y
c
+
z
a
z
b
j
−
z
c
x
a
x
b
+
y
a
y
b
+
z
a
z
b
k
which contains the following dot products:
a
×
b
×
c
=
x
b
a
·
c
i
+
y
b
a
·
c
j
+
z
b
a
·
c
k
−
x
c
a
·
b
i
−
y
c
a
·
b
j
−
z
c
a
·
b
k
(B.3)
Simplifying Eq. (B.3) gives
a
×
b
×
c
=
a
·
c
x
b
i
+
y
b
j
+
z
b
k
−
a
·
b
x
c
i
+
y
c
j
+
z
c
k
which can be written as
a
×
b
×
c
=
a
·
c
b
−
a
·
b
c
(B.4)
