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and the line's direction vector is
a
:
a
=
i
+
j
therefore
n
=
√
3and
a
=
√
2
and
α
=cos
−
1
2
=35
.
26
◦
√
6
10.8.4 Intersection of a Line with a Plane
Given a line and a plane, they will either intersect or are parallel. Either way,
both conditions can be found using some simple vector analysis, as shown in
Figure 10.43.
The objective is to identify a point
P
that is on the line and the plane.
Let the plane equation be
ax
+
by
+
cz
+
d
=0
where
n
=
a
i
+
b
j
+
c
k
P
is a point on the plane with position vector
p
=
x
i
+
y
j
+
z
k
therefore
n
·
p
+
d
=0
Let the line equation be
p
=
t
+
λ
v
where
t
=
x
T
i
+
y
T
j
+
z
T
k
Y
n
v
P
T
p
t
Z
X
Fig. 10.43.
The vectors required to determine whether a line and plane intersect.
