Graphics Reference
In-Depth Information
6.2.8 Scalar Product
We could multiply two vectors r and s by using the product of their magni-
tudes:
. Although this is a valid operation, it does not get us any-
where because it ignores the orientation of the vectors, which is one of their
important features. The concept, however, is readily developed into a useful
operation by including the angle between the vectors.
Figure 6.7 shows two vectors r and s that have been drawn, for conve-
nience, 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
s
·
r =
||
s
|| · ||
r
||
cos( β )
(6.18)
The dot symbol '
' is used to represent scalar multiplication, to distinguish
it from the vector product, which, we will discover, employs a '
·
' symbol.
Because of this symbol, the scalar product is often referred to as the dot
product .
So far we have only defined what we mean by the dot product. We now
need to find out how to compute it. Fortunately, everything is in place to
perform this task. To begin with, we define two Cartesian vectors r and s ,
and proceed to multiply them together using the dot product definition:
×
r = a i + b j + c k
(6.19)
s = d i + e j + f k
(6.20)
therefore
r · s =( a i + b j + c k ) · ( d i + e j + f k )
= a i ·
( d i + e j + f k )+
b j ·
( d i + e j + f k )+
c k ·
( d i + e j + f k )
Y
r
b
s
X
Z
Fig. 6.7. The projection of r on s creates the basis for the scaler product.
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