Graphics Reference
In-Depth Information
r
·
s
=
ad
(
i
k
)+
bd
(
j
·
i
)+
be
(
j
·
j
)+
bf
(
j
·
k
)+
cd
(
k
·
i
)+
ce
(
k
·
j
)+
cf
(
k
·
k
)
·
i
)+
ae
(
i
·
j
)+
af
(
i
·
(6.21)
Before we proceed any further, we can see that we have created various dot
product terms such as (
i
·
i
), (
j
·
j
), (
k
·
k
), etc. These terms can be di-
vided into two groups: those that involve the same unit vector, and those that
reference different unit vectors.
Using the definition of the dot product, terms such as (
i
·
i
), (
j
·
j
)and
(
k
·
k
) = 1, because the angle between
i
and
i
,
j
and
j
,or
k
and
k
is 0
◦
;and
cos(0
◦
) = 1. But because the other vector combinations are separated by 90
◦
,
and cos(90
◦
) = 0, all remaining terms collapse to zero. Bearing in mind that
the magnitude of a unit vector is 1, we can write
||
s
|| · ||
r
||
cos(
β
)=
ad
+
be
+
cf
(6.22)
This result confirms that the dot product is indeed a scalar quantity. Now
let's see how it works in practice.
6.2.9 Example of the Dot Product
To find the angle between two vectors
r
and
s
,
⎡
⎤
⎡
⎤
2
5
6
10
⎣
⎦
⎣
⎦
r
=
−
3
4
and
s
=
=
2
2
+(
||
r
||
−
3)
2
+4
2
=5
.
385
=
5
2
+6
2
+10
2
=12
.
689
||
s
||
Therefore
||
s
|| · ||
r
||
cos(
β
)=2
×
5+(
−
3)
×
6+4
×
10 = 32
12
.
689
×
5
.
385
×
cos(
β
)=32
32
12
.
689
cos(
β
)=
=0
.
468
×
5
.
385
β
=cos
−
1
(0
.
468) = 62
.
1
◦
The angle between the two vectors is 62
.
1
◦
.
It is worth pointing out at this stage that the angle returned by the dot
product ranges between 0
◦
and 180
◦
. This is because, as the angle between
two vectors increases beyond 180
◦
, the returned angle
β
is always the smallest
angle associated with the geometry.
