Game Development Reference
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 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
Scaling and adding together an arbitrary number of vectors is called a linear
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
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 .
This is known as an affine combination . We can express an affine combination as
x = 1
x n +
a i x i