Graphics Reference
In-Depth Information
2.9 Vector products
We now come to vector products, which originally took mathematicians of the day some time
to recognise and define. What was unusual was that there were two ways of multiplying vectors
together: one that produced a scalar quantity and another that created a new vector.
The product of two scalars is straightforward. For example, 3
12 —
order has no impact on the final result. However, there are two interpretations of this product:
one is simply the fact that 3 lots of 4 make 12; the other is that 3
×
4
=
12 and 4
×
3
=
4 can be thought of as an
area, 12 units in size. The product of two vectors is not quite so simple, but there are similarities
with scalar multiplication. For those readers who wish to know how the two products were
discovered, see Michael Crowe's excellent topic,
A History of Vector Analysis
[Crowe, 67].
×
2.9.1 Scalar product
Before we define this product, it's worth thinking about what we would have predicted the
scalar product could have been based upon our knowledge of scalar multiplication. For example,
given the following vectors:
a
=
x
a
i
+
y
a
j
+
z
a
k
b
=
x
b
i
+
y
b
j
+
z
b
k
there is a temptation to multiply the corresponding vector components together:
x
a
x
b
a
y
b
a
z
b
But what do we do with these terms? Well, why not add them together? If we do, we get
a
·
b
=
x
a
x
b
+
y
a
y
b
+
z
a
z
b
Well, it just so happens that this
is
the definition of the scalar product!
An alternative approach would be to multiply their lengths together:
·
, which is a
scalar. However, this ignores the orientation of the two vectors, which might tempt us to suggest
one of the following extensions:
a
b
a
·
b
cos
a
·
b
sin
a
·
b
where is the separating angle between the vectors.
These all seem reasonable, and in mathematics there are no rules preventing anyone from
defining some new formula or technique. What is important is that it integrates with the rest of
mathematics. Well, as we are about to discover, the first two suggestions are extremely useful,
but the third has no application.
The
scalar product
of two vectors
a
·
b
is defined as
a
·
b
=
a
·
b
cos
(2.3)
