Geoscience Reference
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Fig. 7.1 Direction angles
r
0
ﾼ~
~
r
By applying the expression of their components, the above equation becomes:
X
0
i
0
þ
Y
0
j
0
þ
Z
0
k
0
ﾼ
X
i
Y
j
Zk ,
þ
þ
ð
7
:
1
Þ
where
i
0
,
j
0
, and k
0
are the basic unit vectors of O
X
0
Y
0
Z
0
, while
i ,
j , and k are the
‐
XYZ; X
0
, Y
0
, Z
0
and X, Y, Z are the components of
r
0
,
basic unit vectors of O
~
r and
~
‐
respectively.
Taking the dot product (scalar product) of both sides of (
7.1
) with
i
0
and then
with
j
0
and k
0
, respectively, yields:
k ,
X
0
X
i
0
i
Y
i
0
j
Z
i
0
ﾼ
þ
þ
k ,
Y
0
X
j
0
i
Y
j
0
j
Z
j
0
ﾼ
þ
þ
and
Z
0
Xk
0
i
Yk
0
j
Zk
0
k
ﾼ
þ
þ
:
From the definition of the dot product of two vectors, we have:
cos
i
0
;
i
ﾼ
i
0
i
ﾼ
cos
α
1
,
cos
i
0
;
j
ﾼ
i
0
j
ﾼ
cos
ʲ
1
,
cos
i
0
;k
i
0
k
ﾼ
ﾼ
cos
ʳ
1
,
⋮
k
0
k
0
;k
k
ﾼ
cos
ﾼ
cos
ʳ
3
:
Hence, the above equation can be rewritten as:
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