Graphics Reference
In-Depth Information
Example 2
Prove that two vectors are perpendicular, given that
a
=
i
+
3
j
−
2
k
and
b
=
4
i
+
2
j
+
5
k
Therefore,
a
·
b
=
i
+
3
j
−
2
k
·
4
i
+
2
j
+
5
k
=
0
Because
> 0 and the dot product is zero, it can only mean that the separating
angle of the vectors is 90
, whose cosine is zero.
Having looked at two examples, let's return to the geometry behind the dot product.
Figure 2.16 shows that
−
AB
a
> 0 and
b
−
AB
·
−
AC
−
AB
·
−
AC
·
−
AC
=
cos
=
D
C
B
AD
AB
AC
C
′
θ
AC
′
θ
′
A
Figure 2.17.
But Fig. 2.17 introduces a point D such that the line
−−−→
C
CD is perpendicular to
−
AB. Surely, the
projection of
−
AD on
−
AB is
−
AC
, which equals the projection of
−
AC on
−
AB? In fact, all points on
a line perpendicular to another line will have a common projection.
Therefore,
−
AB
−
AD
−
AB
−
AC
−
AB
·
−
AD
cos
=
=
This geometric configuration often arises when solving problems. For example, with reference
to Fig. 2.18, we can state directly that
p
·
v
=
q
·
v
simply because
p
and
q
have identical projections on
v
.
