Geology Reference
In-Depth Information
Fig. 12.3.
Geometry of the dot product
and cross product.
Θ
is the
angle between
v
1
and
v
2
;
v
3
is
perpendicular to the plane
of
v
1
and
v
2
. The
shaded plane
is perpendicular to
v
1
and
the
ruled plane
is perpen-
dicular to
v
2
.
(12.26)
In Eq. 12.26,
v
3
is the vector perpendicular to both
v
1
and
v
2
. The order of mul-
tiplication changes the direction of
v
3
but not the orientation of the line or its mag-
nitude.
12.4.3
Line of Intersection between Two Planes
The properties of the cross product make it suitable for solving a number of problems
in structural analysis. For example, two dip domains can be defined by their poles (or
their dip vectors) and the cross product of these vectors will give the orientation of the
hinge line between them. If the pole to a plane is defined by the azimuth (
)
of a vector pointing in the dip direction, the orientation of this vector in terms of di-
rection cosines is given by Eqs. 12.13, which are
θ
) and dip (
δ
l
=cos
α
p
= -sin
δ
sin
θ
,
(12.27a)
m
=cos
β
p
= -sin
δ
cos
θ
,
(12.27b)
n
=cos
γ
p
=-cos
δ
.
(12.27c)
The subscript “p” = the pole to a plane. Let the subscript “1” = the first do-
main dip, 2 = the second domain dip, and h = the hinge line, then substitute Eqs. 12.27
into 12.26 to obtain a vector parallel to the hinge line in terms of the direction co-
sines:
cos
α
=sin
δ
1
cos
θ
1
cos
δ
2
-cos
δ
1
sin
δ
2
cos
θ
2
,
(12.28a)
cos
β
=cos
δ
1
sin
δ
2
sin
θ
2
-sin
δ
1
sin
θ
1
cos
δ
2
,
(12.28b)
cos
γ
=sin
δ
1
sin
θ
1
sin
δ
2
cos
θ
2
-sin
δ
1
cos
θ
1
sin
δ
2
sin
θ
2
.
(12.28c)
