Game Development Reference

In-Depth Information

Figure 1.2.
Vector scaling and addition.

shows the relationship between points and vectors—in this case, the vector is

acting as the difference between two points. Algebraically, this is

v
=
x
1
−

x
0

or

x
1
=
x
0
+
v
.

In general, vectors can be scaled and added. Scaling (multiplying by a single

factor, or scalar) changes the length of a vector. If the scalar is negative, it can

also change the direction of the vector. Adding two vectors together creates a new

vector that points from the tail of one to the head of another (see
Figure 1.2
).

Scaling and adding together an arbitrary number of vectors is called a
linear

combination
:

v
=

i

a
i
v
i
.

A set of vectors
v
is
linearly dependent
if one of the vectors in
S
can be repre-

sented as the linear combination of other members in
S
.Otherwise,itisa
linearly

independent
set.

Points cannot be generally scaled or added. They can only be subtracted to

create a vector or combined in a linear combination, where

a
i
=1
.

i

This is known as an
affine combination
. We can express an affine combination as

follows:

x
=
1

x
n
+

n−
1

n−
1

−

a
i

a
i
x
i

i

i