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and the sphere misses the plane because
> 5.
If the plane is represented parametrically as shown in Fig. 6.22, then using two vectors
a
and
b
gives
p
=
t
+
a
+
b
where and are scalars, we can proceed as follows.
Y
P
CP
C
c
p
b
a
n
O
d
y
C
t
T
Z
x
C
z
C
X
Figure 6.22.
We find the plane's normal vector
n
using the cross product
=
×
n
a
b
from which
n
ˆ
=
n
n
Therefore,
n
ˆ
·
t
=
d
which can be substituted in Eq. (6.36):
=ˆ
n
·
t
−ˆ
n
·
c
=ˆ
n
·
t
−
c
n
=
·
t
−
c
n
and
c
a
×
b
=
·
t
−
a
×
b
Let's test this technique with a simple example shown in Fig. 6.23.
