Graphics Reference
In-Depth Information
Appendix B
B.1 Vector triple product
×
×
=
·
−
·
×
×
=
In this appendix we will prove that
a
b
c
a
c
b
a
b
c
and
a
b
c
·
c
a
, which are both known as
vector triple products
.
In Section 2.10.1 we proved that
a
−
·
a
c
b
b
×
×
b
c
resides in the plane containing
b
and
c
. Therefore,
×
×
a
b
c
must be linearly related to
b
and
c
. Thus,
×
×
=
+
a
b
c
b
c
(B.1)
c
and force the algebra
into the form of the RHS of Eq. (B.1). At the same time, we must look out for any hidden dot
products. Let's begin the expansion: given
a
×
b
×
where and (delta) are scalars. Now we have to expand
a
=
+
+
x
a
i
y
a
j
z
a
k
, we have
b
=
x
b
i
+
y
b
j
+
z
b
k
c
=
x
c
i
+
y
c
j
+
z
c
k
Then
i j k
x
b
y
b
z
b
x
c
y
c
z
c
y
b
z
b
y
c
z
c
z
b
x
b
z
c
x
c
x
b
y
b
x
c
y
c
b
×
c
=
=
i
+
j
+
k
and
b
×
c
=
y
b
z
c
−
y
c
z
b
i
+
z
b
x
c
−
z
c
x
b
j
+
x
b
y
c
−
x
c
y
b
k
Therefore,
i
j
k
a
×
b
×
c
=
x
a
y
a
z
a
y
b
z
c
−
y
c
z
b
z
b
x
c
−
z
c
x
b
x
b
y
c
−
x
c
y
b
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