Graphics Reference
In-Depth Information
line segments
to explain how vectors work. The second approach provides a rapid entry into
the subject and will be sufficient for most readers. Appendix A provides a formal description
of vectors for those readers who desire it.
Before vector notation was developed, problems involving forces were resolved using line
segments, and forces were added together using the parallelogram rule shown in Fig. 1.1.
force
A
+
B
force
B
force
A
Figure 1.1.
The idea of using line segments to represent vectors has dominated the evolution of vector
analysis, although it must be noted that some mathematicians have no need of such visual aids
when they enter multi-dimensional worlds of abstract vector spaces! Fortunately for us, all of the
problems considered in the following chapters are based upon simple two- or three-dimensional
line segments.
A line segment is a perfect graphic for representing a vector, because its length represents
magnitude, and its orientation represents direction. Figure 1.2 shows three identical vectors. They
are identical because their length and orientation are equal, and their positions are irrelevant.
Figure 1.2.
The line segments in Fig. 1.2 have arrow heads, which define the vector's direction. Without
the arrow the line segment could be pointing in either direction. Thus, a directed line segment
is required to provide an unambiguous description of a vector.
Let us develop a notation for these line segments, one that will reflect the axioms that
underpin vector algebra.
Figure 1.3(a) shows two points, A and B, connected by a line with an arrow pointing in the
direction from A to B. This directed line segment is labeled
−
AB, where the arrow in the diagram
confirms the direction from A to B, and the arrow on top of AB reminds us that it is a vector
quantity.
