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0
−
0
00
01
10
01
z
0
=
=0
1
Therefore the line equation is
p
=
λ
n
3
.
where
n
3
=
j
, which is the
y
-axis.
10.8.1 Intersection of Three Planes
Three mutually intersecting planes will intersect at a point as shown in Fig-
ure 10.37, and we can find this point by using a similar strategy to the one
used in two intersecting planes by creating three simultaneous plane equations
using determinants.
Figure 10.37 shows three planes intersecting at the point
P
(
x
,
y
,
z
).
The three planes can be defined by the following equations:
a
1
x
+
b
1
y
+
c
1
z
+
d
1
=0
a
2
x
+
b
2
y
+
c
2
z
+
d
2
=0
a
3
x
+
b
3
y
+
c
3
z
+
d
3
=0
which means that they can be rewritten as
⎡
⎤
⎡
⎤
⎡
⎤
−
d
1
−
a
1
b
1
c
1
x
y
z
⎣
⎦
=
⎣
⎦
·
⎣
⎦
d
2
−
a
2
b
2
c
2
d
3
a
3
b
3
c
3
or
⎡
⎤
⎡
⎤
⎡
⎤
d
1
d
2
d
3
a
1
b
1
c
1
x
y
z
⎣
⎦
=
⎣
⎦
·
⎣
⎦
−
a
2
b
2
c
2
a
3
b
3
c
3
or in determinant form:
x
y
z
=
−
1
DET
=
=
d
1
b
1
c
1
a
1
d
1
c
1
a
1
b
1
d
1
d
2
b
2
c
2
a
2
d
2
c
2
a
2
b
2
d
2
d
3
b
3
c
3
a
3
d
3
c
3
a
3
b
3
d
3
Y
P
Z
X
Fig. 10.37.
Three mutually intersecting planes.
