Graphics Reference
In-Depth Information
The multiplication and division of two vectors are not so obvious. To begin with, dividing one
vector by another does not seem to mean anything and has no definition, whereas vectors can
be multiplied together in two ways, and we explore these products later.
The laws associated with scalars are simple and very familiar. Fortunately, the same laws
apply to vectors; the only difference between the two systems is in the product of two vectors,
which have still to be covered. Table 2.2 gives examples of the two systems.
Table 2.2
Law
Scalar algebra
Vector algebra
Commutative law for addition
a
+
b
=
b
+
a
a
+
b
=
b
+
a
Associative law for addition
a
+
b
+
c
=
a
+
b
+
c
a
+
b
+
c
=
a
+
b
+
c
Commutative law for multiplication
ab
=
ba
a
b
=
b
a
Associative law for multiplication
abc
=
abc
ab
c
=
ab
c
Distributive law for multiplication
a
+
bc
=
ac
+
bc
a
+
b
c
=
a
c
+
b
c
ab
+
c
=
ab
+
ac
a
b
+
c
=
a
b
+
a
c
2.6 Unit vectors
A
unit vector
has a length of 1. Unit vectors greatly simplify problem solving; therefore, we
should understand how they are created.
For ex
ample, if
v
=
xy
T
and is a unit vector, we are implying that
v
=
1, i.e.,
x
2
n
, where the circumflex reminds us of the vector's unit
length. We already know that a vector's length can be controlled by a scaling factor. Therefore,
any vector has to be some multiple of a unit vector:
+
y
2
=
1. It is normally written as
ˆ
=
ˆ
v
v
But, surely,
=
v
. Therefore, we can write
v
=
v
ˆ
v
which leads us to
v
v
ˆ
=
v
For example, given
3
4
v
=
we can make it a unit vector by dividing its components by
v
, which, in this case, is
√
3
2
+
4
2
=
5
i.e.,
06
08
v
ˆ
=
