Suppose that the quadrilateral ABCD in Fig. 13 is a parallelogram. It follows that
the opposite sides of ABCD can be represented by the
*same* vectors, and : this merely indicates that these sides are of
equal length and are parallel (*i.e.*, they point in the same direction). Note that
Fig. 13 illustrates an important point regarding vectors. Although vectors possess
both a magnitude (length) and a direction, they possess no intrinsic position information.
Thus, since sides and are parallel and of equal length, they can be represented
by the *same* vector , despite the fact that they are in different places on the
diagram.

The diagonal in Fig. 13 can be represented vectorially as
.
Likewise, the diagonal can be written
.
The displacement (say) of the centroid from point can be written in one
of two different ways:

Equation (38) is interpreted as follows: in order to get from point to point , first move to point (along vector ), then move along diagonal (along vector ) for an unknown fraction of its length. Equation (39) is interpreted as follows: in order to get from point to point , first move to point (along vector ), then move to point (along vector ), finally move along diagonal (along vector ) for an unknown fraction of its length. Since represents the

(40) |

(41) | |||

(42) |

It follows that . In other words, the centroid is located at the halfway points of diagonals and :