Geology Reference
In-Depth Information
C
=
x
1
y
2
+
y
1
x
3
+
x
2
y
3
-
y
2
x
3
-
y
3
x
1
-
y
1
x
2
,
(12.16c)
D
=
z
1
y
2
x
3
+
z
2
y
3
x
1
+
z
3
y
1
x
2
-
x
1
y
2
z
3
-
y
1
z
2
x
3
-
z
1
x
2
y
3
.
(12.16d)
The direction cosines of the pole to this plane are (Eves 1984)
cos
α
p
=
A
/
E
,
(12.17a)
cos
β
p
=
B
/
E
,
(12.17b)
cos
γ
p
=
C
/
E
,
(12.17c)
where
E
=(
A
2
+
B
2
+
C
2
)
1/2
,
(12.18)
and the sign of
E
=-sign
D
if
D
0; = sign
B
if
C
=
D
=0.
The direction cosines of the pole may be converted to the direction cosines of the
dip vector. If cos
≠
0; = sign
C
if
D
=0 and
C
≠
γ
p
is negative, the pole points downward. Reverse the signs on all
three direction cosines to obtain the upward direction. If cos
γ
p
is positive, the pole
points upward, and the dip azimuth is the same as that of the pole and the dip amount
is equal to 90° plus the angle between the pole and the
z
axis:
cos
α
=
A
/
E
,
(12.19a)
cos
=
B
/
E
,
(12.19b)
β
cos
γ
= cos (90 + arccos (
C
/
E
)) .
(12.19c)
To find the azimuth and plunge of the dip direction, use Eqs. 12.4 and 12.5 and
Table 12.1 or the Excel equation 12.6.
12.4
Vector Geometry of Lines and Planes
Vector geometry (Thomas 1960) is an efficient method for the analytical computation
of the relationships between lines and planes. A 3-D vector has the form
v
=
l
i
+
m
j
+
n
k
,
(12.20)
where
i
,
j
,
k
= unit vectors parallel to the
x
,
y
,
z
axes, respectively (Fig. 12.2) and the
coefficients are the direction cosines
l
,
m
,
n
of the line with respect to the correspond-
ing axes. The vector
v
is a unit vector if its length is equal to one, and gives the orien-
tation of a line. The length of the vector is
|
l
i
+
m
j
+
n
k
|=(
l
2
+
m
2
+
n
2
)
1/2
,
(12.21)
