Graphics Reference
In-Depth Information
units long aligned with the
x
-axis is simply 10
i
, and a vector 20 units long
aligned with the
z
-axis is 20
k
. By employing the rules of vector addition and
subtraction, we can compose a vector
r
by adding three
Cartesian
vectors as
follows:
r
=
a
i
+
b
j
+
c
k
(6.12)
This is equivalent to writing
r
as
⎡
⎤
a
b
c
⎣
⎦
r
=
(6.13)
which means that the magnitude of
r
is readily computed as
=
a
2
+
b
2
+
c
2
||
r
||
(6.14)
Any pair of Cartesian vectors such as
r
and
s
can be combined as follows:
r
=
a
i
+
b
j
+
c
k
(6.15)
s
=
d
i
+
e
j
+
f
k
(6.16)
r
±
s
=(
a
±
d
)
i
+(
b
±
e
)
j
+(
c
±
f
)
k
(6.17)
For example, given
r
=2
i
+3
j
+4
k
and
s
=5
i
+6
j
+7
k
then
r
+
s
=7
i
+9
j
+11
k
and
=
7
2
+9
2
+11
2
=
√
251
||
r
+
s
||
= 15
.
84
6.2.7 Vector Multiplication
Although vector addition and subtraction are useful in resolving various prob-
lems, vector multiplication provides some powerful ways of computing angles
and surface orientations.
The multiplication of two scalars is very familiar: for example, 6
×
6 = 42. We often visualize this operation, as a rectangular area where 6 and
7 are the dimensions of a rectangle's sides, and 42 is the area. However, when
we consider the multiplication of vectors we are basically multiplying two 3D
lines together, which is not an easy operation to visualize.
Mathematicians have discovered that there are two ways to multiply vec-
tors together: one gives rise to a scalar result and the other a vector result.
We will start with the
scalar product
.
×
7or7
