LINES and PLANES
Maths21a, O. Knill
LINES. A point
P
and a vector
vectorv
define a line
L
. It is the set of points
L
=
{
P
+
tvectorv
, where
t
is a real number
}
The line contains the point
P
and points into the direction
vectorv
.
EXAMPLE.
L
=
{
(
x, y, z
) = (1
,
1
,
2) +
t
(2
,
4
,
6)
}
.
This description is called the
parametric equation
for the line.
EQUATIONS OF LINE. We can write (
x, y, z
) = (1
,
1
,
2) +
t
(2
,
4
,
6) so that
x
= 1 + 2
t, y
= 1 + 4
t, z
= 2 + 6
t
. If
we solve the first equation for
t
and plug it into the other equations, we get
y
= 1 + (2
x

2)
, z
= 2 + 3(2
x

2).
We can therefore describe the line also as
L
=
{
(
x, y, z
)

y
= 2
x

1
, z
= 6
x

4
}
SYMMETRIC EQUATION. The line
vector
r
=
P
+
tvectorv
with
P
= (
x
0
, y
0
, z
0
) and
vectorv
= (
a, b, c
) satisfies the
symmetric
equations
x

x
0
a
=
y

y
0
b
=
z

z
0
c
(every expression is equal to
t
).
PROBLEM. Find the equations for the line through the points
P
= (0
,
1
,
1) and
Q
= (2
,
3
,
4).
SOLUTION. The parametric equations are (
x, y, z
) = (0
,
1
,
1) +
t
(2
,
2
,
3) or
x
= 2
t, y
= 1 + 2
t, z
= 1 + 3
t
.
Solving each equation for
t
gives the symmetric equations
x/
2 = (
y

1)
/
2 = (
z

1)
/
3.
PLANES. A point
P
and two vectors
vectorv, vectorw
define a plane Σ. It is the set of points
Σ =
{
P
+
tvectorv
+
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 Spring '08
 FABER
 Equations, Parametric equation

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