Game Physics Engine Development, 2nd Edition

178 Chapter 9 The M athematics of Rotations

which we can verify with the matrix multiplication,

⎡

⎢

⎣

abc

def

ghi

⎤

⎥

⎦

⎡

⎢

⎣

1

0

0

⎤

⎥

⎦

=

⎡

⎢

⎣

a ×1 +b × 0 + c × 0

d ×1 +e × 0 +f × 0

g × 1 +h ×0 + i ×0

⎤

⎥

⎦

=

⎡

⎢

⎣

a

d

g

⎤

⎥

⎦

and so on for the other two axes. When I introduced vectors , I mentioned that their

three components could be thought of as a position along three axes. The x compo-

nent is the distance along the X axis and so on. We could write the vector as

v =

⎡

⎢

⎣

x

y

z

⎤

⎥

⎦

=x

⎡

⎢

⎣

1

0

0

⎤

⎥

⎦

+y

⎡

⎢

⎣

0

1

0

⎤

⎥

⎦

+z

⎡

⎢

⎣

0

0

1

⎤

⎥

⎦

In other words, a vector is made up of some proportion of each basic axis.

If the three ax es move under a transformation, then the new location o f the vector

will be determined in the same way as before. The axes will have moved but the new

vector will still combine them