Graphics Reference
In-Depth Information
with the
x
-axis is 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 summing three
Cartesian unit vector
as follows:
r
=
a
i
+
b
j
+
c
k
which is equivalent to writing
r
as
⎡
⎤
a
b
c
⎣
⎦
=
r
and means that the length of
r
is computed as
a
2
b
2
c
2
.
|
r
|=
+
+
Any pair of Cartesian vectors such as
r
and
s
are combined as follows
=
+
+
r
a
i
b
j
c
k
s
=
d
i
+
e
j
+
f
k
r
±
s
=
(a
±
d)
i
+
(b
±
e)
j
+
(c
±
f )
k
.
For example:
r
=
2
i
+
3
j
+
4
k
s
=
5
i
+
6
j
+
7
k
r
+
s
=
7
i
+
9
j
+
11
k
.
3.10 Scalar Product
The mathematicians who defined the structure of vector analysis provided two ways
to multiply vectors together: one gives rise to a scalar result and the other a vector
result. For example, we could multiply two vectors
r
and
s
by using the product of
their magnitudes:
. Although this is a valid operation it ignores the orientation
of the vectors, which is one of their important features. The idea, however, is readily
developed into a useful operation by including the angle between the vectors.
Figure
3.3
shows two vectors
r
and
s
that have been drawn, for convenience,
such that their tails touch. Taking
s
as the reference vector - which is an arbitrary
choice - we compute the projection of
r
on
s
, which takes into account their relative
orientation. The length of
r
on
s
is
|
r
||
s
|
|
|
r
cos
β
. We can now multiply the magnitude of
|
||
|
s
by the projected length of
r
:
s
r
cos
β
.
This scalar product is written
r
·
s
=|
r
||
s
|
cos
β.
(3.1)
The dot symbol '
' is used to denote a scalar multiplication, which is why the product
is often referred to as the
dot product
. We now need to discover how to compute it.
·
