Intersection of a Line and a Plane in 3-Space Activity
1.
a.
Line is intersected at the point (3, -7,5) so:
Line should not be parallel to the plane
(3, -7,5) lies on the plane
(
a
^
i
+
b
^
j
+
c
^
k
)
.
(
−
2
^
i
+
4
^
j
+
7
^
k
)
≠
0
−
2
a
+
4
b
+
7
c ≠
0
3
a
−
7
b
−
5
c
=
d
Equationof planeis
ax
+
by
+
cz
=
3
a
−
7
b
−
5
c withcondition of
2
a
−
4
b
−
7
c≠
0

b.
Plane ins not intersected by the line
Line is parallel to plane
Normal vector of the plane is perpendicular to line vector
−
2
a
+
4
b
+
7
c
=
0
ax
+
by
+
cz
=
disequationof plane with condition
−
2
a
+
4
b
+
7
c
=
0
c.
Plane that the line lies on
Point (1, -3,2) lies on the plane
.
(
−
2
^
i
+
4
^
j
+
7
^
k
)
is perpendicular to normal vector
a
−
3
b
+
2
c
=
d
−
2
a
+
4
b
+
7
c
=
0
Equationof planeisax
+
by
+
cz
=
a
−
3
b
+
2
cwith condition
−
2
a
+
4
b
+
7
c
=
0

2.
a.
Plane intersects x-axis at