Geology Reference
In-Depth Information
The direction cosines must be normalized to ensure that the sum of their squares
equals 1, which is done by dividing through by
N
c
where
2
+cos
2
+cos
2
)
1/2
.
N
c
= (cos
α
β
γ
(12.29)
The final equation for the line of intersection between two planes is
cos
α
h
= (cos
α
)/
N
c
,
(12.30a)
cos
β
h
= (cos
β
)/
N
c
,
(12.30b)
cos
γ
h
=(cos
γ
)/
N
c
.
(12.30c)
The azimuth and dip of the line of intersection are given by Eqs. 12.4, 12.5 and Table 12.1.
12.4.4
Plane Bisecting Two Planes
The axial surface of a constant thickness fold hinge is a plane that bisects the angle be-
tween the two adjacent bedding planes and can be found as a vector sum. A vector sum
is the diagonal of the parallelogram formed by two vectors. If the two vectors forming the
sum have the same lengths, as do unit vectors, then the diagonal bisects the angle between
them (Fig. 12.4). The sum of two vectors is the sum of their corresponding components:
v
1
+
v
2
=(
l
1
+
l
2
)
i
+(
m
1
+
m
2
)
j
+(
n
1
+
n
2
)
k
.
(12.31)
The unit vector that bisects the angle between the two vectors
v
1
and
v
2
is
v
3
=(1/
N
b
)(
l
1
+
l
2
)
i
+(1/
N
b
)(
m
1
+
m
2
)
j
+(1/
N
b
)(
n
1
+
n
2
)
k
,
(12.32)
where
N
b
=((
l
1
+
l
2
)
2
+(
m
1
+
m
2
)
2
+(
n
1
+
n
2
)
2
)
1/2
(12.33)
serves to normalize the length to make the resultant a unit vector. Depending on the
directions of
v
1
and
v
2
, this vector could bisect either the acute angle or the obtuse
Fig. 12.4.
Bisecting vector (
b
) of the
angle between two vectors in
the plane of the two vectors
