Graphics Reference
In-Depth Information
other. For example, Fig. 1.11 shows triangle ABC constructed from three non-collinear vectors:
r
=
−
AB
s
=
−
BC, and
r
=
−
AC. Superimposed is another triangle ADE with a
r
=
−
AD b
s
=
−
DE,
+
s
=
−
AE, where a and b are scalars. The vectors a
r
and b
s
are also non-collinear,
because
−
AD is parallel to
−
AB, and
−
DE is parallel to
−
BC. This is written formally as
−
AD
+
and a
r
b
s
−
AB and
−
BC, where the symbol
−
DE
means
parallel to
.
E
C
a
r
+
b
s
b
s
r
+
s
s
D
a
r
B
r
A
Figure 1.11.
Examine Fig. 1.11 very carefully and note that the route
−
AE is unique. For no matter how we
scale the vectors
r
and
s
, and add them together, their sum will always produce a different
resultant vector. If we want to move from A to E, only one pair of scalars exists to scale
r
and
s
. This means that if ever we encounter a statement such as
a
r
+
b
s
=
c
r
+
d
s
where
r
and
s
are non-collinear, it can only mean one thing: a
d. It is worth
dwelling on this point until you really appreciate why this is so. This condition only applies
to non-collinear vectors. However, such vector combinations are very common, and play an
important role in problem solving.
We now know enough vector algebra to solve some simple geometric problems. So, consider
proving that the opposite sides of a parallelogram are equal. We begin with the parallelogram
ABCD shown in Fig. 1.12, where
r
=
c and b
=
=
−
AB and
s
=
−
AD. But as AB
DC and AD
BC,wecan
conjecture that
−
DC
u
r
and
−
BC
=
=
v
s
, where u and v are scalars.
u
r
D
C
s
v
s
AC
A
B
r
Figure 1.12.
