Graphics Reference
In-Depth Information
2.9.4 The algebra of vector products
Before proceeding with triple products, we need to consolidate the algebraic rules associated with
the dot and cross products. Two laws need to be considered: the commutative and distributive
laws, but we also need to confirm the role of scalars in these products.
Commutative law a
·
b
=
b
·
a
As
a
·
b
=
a
b
cos
(2.13)
the commutative law of scalar multiplication permits us to write Eq. (2.13) as
a
·
b
=
b
a
cos
=
b
·
a
Therefore,
a
·
b
=
b
·
a
Distributive law a
·
b
+
c
=
a
·
b
+
a
·
c
Given
a
=
x
a
i
+
y
a
j
+
z
a
k
b
=
x
b
i
+
y
b
j
+
z
b
k
c
=
x
c
i
+
y
c
j
+
z
c
k
then
a
·
b
+
c
=
x
a
i
+
y
a
j
+
z
a
k
·
x
b
i
+
y
b
j
+
z
b
k
+
x
c
i
+
y
c
j
+
z
c
k
a
·
b
+
c
=
x
a
i
+
y
a
j
+
z
a
k
·
x
b
+
x
c
i
+
y
b
+
y
c
j
+
z
b
+
z
c
k
a
·
b
+
c
=
x
a
x
b
+
x
c
+
y
a
y
b
+
y
c
+
z
a
z
b
+
z
c
a
·
b
+
c
=
x
a
x
b
+
x
a
x
c
+
y
a
y
b
+
y
a
y
c
+
z
a
z
b
+
z
a
z
c
=
a
·
b
+
a
·
c
Therefore,
a
·
b
+
c
=
a
·
b
+
a
·
c
Commutative law a
×
b
=
b
×
a
The commutative law does not hold for the vector product.
