Game Development Reference
In-Depth Information
omit the multiplication symbol when multiplying two scalars or a scalar
and a vector, we must not omit the dot symbol when performing a vector
dot product. If you ever see two vectors placed side-by-side with no symbol
in between, interpret this according to the rules of matrix multiplication,
which we discuss in Chapter 4.
7
The dot product of two vectors is the sum of the products of corre-
sponding components, resulting in a scalar:
2
4
3
5
2
4
3
5
= a
1
b
1
+ a
2
b
2
+ + a
n−1
b
n−1
+ a
n
b
n
.
a
1
a
2
.
a
n−1
a
n
b
1
b
2
.
b
n−1
b
n
Vector dot product
This can be expressed succinctly by using the summation notation
n
a
b
=
a
i
b
i
.
Dot product using
summation notation
i=1
Applying these rules to the 2D and 3D cases yields
a
b
= a
x
b
x
+ a
y
b
y
(
a
and
b
are 2D vectors),
2D and 3D dot products
a
b
= a
x
b
x
+ a
y
b
y
+ a
z
b
z
(
a
and
b
are 3D vectors).
Examples of the dot product in 2D and 3D are
4
6
−3
7
= (4)(−3) + (6)(7) = 30,
2
3
2
3
−2
7
0
−1
4
5
4
5
= (3)(0) + (−2)(4) + (7)(−1) = −15.
It is obvious from inspection of the equations that vector dot product
is commutative:
a
b
=
b
a
. More vector algebra laws concerning the dot
product are given in
Section 2.13.
2.11.2 Geometric Interpretation
Now let's discuss the more important aspect of the dot product: what
it means geometrically. It would be di
cult to make too big of a deal
7
One notation you will probably bump up against is treating the dot product as an
ordinary matrix multiplication, denoted by a
T
b if a and b are interpreted as column
vectors, or ab
T
for row vectors. If none of this makes sense, don't worry, we will repeat
it after we learn about matrix multiplication and row and column vectors in Chapter 4.
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