Graphics Reference
In-Depth Information
u
−
v
=
u
+
(
−
v
)
−
(
−
v
)
=
v
v
+
(
−
v
)
=
0
v
+
0
=
0
+
v
=
v
Additionally, given the scalars
r
and
s
, the following identities hold for scalar
multiplication:
r
(
s
v
)
=
(
rs
)
v
(
r
+
s
)
v
=
r
v
+
s
v
s
(
u
+
v
)
=
s
u
+
s
v
1
v
=
v
3.3.3
The Dot Product
The
dot product
(or
scalar product
) of two vectors
u
and
v
is defined as the sum of the
products of their corresponding vector components and is denoted by
u
·
v
. Thus,
u
·
v
=
(
u
1
,
u
2
,
...
,
u
n
)
·
(
v
1
,
v
2
,
...
,
v
n
)
=
u
1
v
1
+
u
2
v
2
+···+
u
n
v
n
.
Note that the dot product is a scalar, not a vector. The dot product of a vector and
itself is the squared length of the vector:
v
1
+
v
2
+···+
v
n
=
2
.
v
·
v
=
v
It is possible to show that the smallest angle
θ
between
u
and
v
satisfies the equation
u
·
v
=
u
v
cos
θ
,
and thus
θ
can be obtained as
cos
−
1
u
·
v
θ
=
.
u
v
As a result, for two nonzero vectors the dot product is positive when
θ
is acute,
θ
negative when
is obtuse, and zero when the vectors are perpendicular (Figure 3.4).
Being able to tell whether the angle between two vectors is acute, obtuse, or at a right
