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In-Depth Information
If you look carefully at Eq. (2.9), you will see features of Eq. (2.6). Now we want a rule similar
to Eq. (2.3) as that used for the scalar product. So let's try the following:
a
×
b
=
a
b
sin
n
ˆ
(2.10)
where the unit vector
n
is perpendicular to both
a
and
b
.
Applying Eq. (2.10) to Eq. (2.9), the terms
i
ˆ
×
i
j
×
j
, and
k
×
k
collapse to zero, because the
separating angle is 0
, and sin 0
=
0. So we are left with
a
×
b
=
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
If we invoke Hamilton's rules for quaternions for vectors:
i
×
j
=
k
j
×
k
=
i
and
k
×
i
=
j
and their reverse forms:
j
×
i
=−
k
k
×
j
=−
i
and
i
×
k
=−
j
we obtain
a
×
b
=
x
a
y
b
k
−
x
a
z
b
j
−
y
a
x
b
k
+
y
a
z
b
i
+
z
a
x
b
j
−
z
a
y
b
i
Reordering the terms gives
a
×
b
=
y
a
z
b
i
−
z
a
y
b
i
+
z
a
x
b
j
−
x
a
z
b
j
+
x
a
y
b
k
−
y
a
x
b
k
and, finally,
a
×
b
=
y
a
z
b
−
z
a
y
b
i
+
z
a
x
b
−
x
a
z
b
j
+
x
a
y
b
−
y
a
x
b
k
which is identical to Eq. (2.6)!
All of this may appear longwinded, but it is necessary to bring to life what happened
historically. A mathematician suddenly thought of a formula for the vector product.
Today, math topics contain statements such as, “The
vector product
is defined as
a
×
b
=
n
=
a
b
sin
n
ˆ
(2.11)
where
is the smaller angle between
a
and
b
and
n
is perpendicular to both
a
and
b
”
Also,
i
j
k
a
×
b
=
x
a
y
a
z
a
(2.12)
x
b
y
b
z
b
