Game Development Reference
In-Depth Information
Note that this gives both the magnitude and direction of the linear, tangential velocity.
Also, be sure to preserve the order of the vectors when taking the cross product—that
is, ω cross r , and not the other way around, which would give the wrong direction for v .
Vector Cross Product
Given any two vectors A and B , the cross product A × B is defined by a third vector C
with a magnitude equal to AB sin θ, where θ is the angle between the two vectors A and
B , as illustrated in the following figure.
C = A × B
C = AB sin θ
The direction of C is determined by the right hand rule. As noted previously, the right
hand rule is a simple trick to help you keep track of vector directions. Assume that A
and B lie in a plane and let an axis of rotation extend perpendicular to this plane through
a point located at the tail of A . Pretend to curl the fingers of your right hand around the
axis of rotation from vector A toward B . Now extend your thumb (as though you are
giving a thumbs up) while keeping your fingers curled around the axis. The direction
that your thumb is pointing indicates the direction of vector C .
In the preceding figure, a parallelogram is formed by A and B (the shaded region). The
area of this parallelogram is the magnitude of C, which is AB sin θ.
There are two equations that you'll need in order to determine the vectors for tangential
and centripetal acceleration. They are:
a n = ω × ( ω × r )
a t = α × r
Another way to look at the quantities v , a n , and a t is that they are the velocity and
acceleration of the particle under consideration, on the rigid body, relative to the point
about which the rigid body is rotating—for example, the body's center-of-mass location.
This is very convenient because, as we said earlier, you'll want to track the motion of
the rigid body as a particle when viewing the big picture without having to worry about