Graphics Reference
In-Depth Information
where i and j equal
√
−
1. But when two such objects are multiplied together, they create terms
such as i
1, but the last
two did present a problem, and there was no obvious solution, so he extended the notation to
·
i, j
·
j, i
·
j, and j
·
i. The first two are no problem as they are equal to
−
q
=
s
+
ai
+
bj
+
ck
where ij, and k equal
√
−
1. When multiplying these objects together, Hamilton ended up with
terms such as i
·
jj
·
k, and k
·
i, which also defied description, until October 16, 1843, when he
thought of the idea that i
·
j
=
k, j
·
k
=
i, and k
·
i
=
j. He also conjectured that j
·
i
=−
kk
·
j
=−
i,
and i
j. On that day, he discovered
quaternions
, which revealed the foundations of the
scalar and vector products.
Hamilton's ij, and k are not the
i
j
, and
k
we have been using, but there are some strange
similarities. His vector product suggested the following manipulation:
·
k
=−
×
=
y
a
z
b
−
+
z
a
x
b
−
+
x
a
y
b
−
a
b
y
b
z
a
i
z
b
x
a
j
x
b
y
a
k
(2.6)
which, clearly, is another vector and turns out to be perpendicular to the plane containing both
a
and
b
. Equation (2.6) can also be expressed in its determinant form:
y
a
z
a
z
a
x
a
x
a
y
a
a
×
b
=
i
+
j
+
k
(2.7)
y
b
z
b
z
b
x
b
x
b
y
b
and is often shown as
i j k
x
a
y
a
z
a
x
b
y
b
z
b
a
×
b
=
(2.8)
which produces the same result as Eq. (2.7) and is easier to remember. Note that the '
' symbol
is used to distinguish this product from the scalar product, and like the scalar product has a
pseudonym: the
cross product
.
Let's approach this product from another way. Given
×
a
=
x
a
i
+
y
a
j
+
z
a
k
and
b
=
x
b
i
+
y
b
j
+
z
b
k
then
a
×
b
=
x
a
i
+
y
a
j
+
z
a
k
×
x
b
i
+
y
b
j
+
z
b
k
which expands to
a
×
b
=
x
a
x
b
i
×
i
+
y
a
y
b
j
×
j
+
z
a
z
b
k
×
k
+
x
a
y
b
i
×
j
+
x
a
z
b
i
×
k
+
y
a
x
b
j
×
i
+
y
a
z
b
j
×
k
+
z
a
x
b
k
×
i
+
z
a
y
b
k
×
j
(2.9)
