Lines and Planes
Lines
Let
l
be a line parallel to vector
v
.
v
=
v
1
i
+
v
2
j
+
v
3
k
Let
P
0
(
a
,
b
,
c
)
be a point on the line and
P
(
x
,
y
,
z
)
be any
other point on the line.
P
0
P
is a vector parallel to the line.
P
0
P
v
so
The parametric equations of a line are:
x
=
a
+
v
1
t
y
=
b
+
v
2
t
z
=
c
+
v
3
t
To find the equation of a line you need a point on the line and a
vector parallel to the line.
ex. Find the equation of the line through P(1, 5, 3) parallel to
vector
v
=
2
, 0
, 1
=
2
i
+
k
.
ex. Find the equation of the line through points P(1, 2, 3) and
Q(3, 6, 9).
ex. Find a vector parallel to the line
x
=
9
-
t
y
=
3
z
=
6
t
ex.
Determine which of the following line equations represent the
same line as
x
=
9
-
t
y
=
3
z
=
6
t
x
1
=
7
+
2
t
x
2
=
11
-
3
t
x
3
=
9
+
t
y
1
=
3
y
2
=
3
y
3
=
3
z
1
=
12
-
12
t
z
2
=
10
+
18
t
z
3
=
6
t
Planes
Let
P
0
(
a
,
b
,
c
)
be a point in the plane.
If P(x, y, z) is any other
point in the plane, then
P
0
P
is a vector that lies in the plane.
Let
n
=
n
1
i
+
n
2
j
+
n
3
k
be a vector perpendicular (normal) to the
plane.
Then
The equation of a plane is
n
1
x
+
n
2
y
+
n
3
z
=
n
⋅
p
where
n
is the normal vector and
p
is the vector formed by the point.
To find the equation of a plane you need a point on the plane and a
vector normal (perpendicular) to the plane.
