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Math 18.02 (Spring 2009): Lecture 5
Planes. Parametric equations of curves and lines
February 12
Reading Material:
From
Simmons
: 17.1 and 17.2.
Last time:
Square Systems. Word problem. How many solutions? Equations of planes
Today:
Planes. Parametric equations of curves and lines
2
Equations of Planes
In this section we recall the
analytic
description of a plane through a point
P
0
= (
x
0
, y
0
, z
0
) and
perpendicular to a vector
~
N
= (
a, b, c
). This plane is described by all points
P
= (
x, y, z
) such that
the vector
→
P
0
P
= (
x

x
0
, y

y
0
, z

z
0
) is perpendicular to the given vector
~
N
:
~
N
⊥
→
P
0
P
⇐⇒
~
N
·
→
P
0
P
= 0
.
Since
~
N
·
→
P
0
P
=
a
(
x

x
0
) +
b
(
y

y
0
) +
c
(
z

z
0
) = 0
,
it follows that the (
analytic
) equation for our plane is
a
(
x

x
0
) +
b
(
y

y
0
) +
c
(
z

z
0
) = 0
.
(2.1)
1
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View Full DocumentExercise 1.
Find a plane containing
P
1
= (1
,
2
,
3)
, P
2
= (1
,
4
,
4)
and
P
3
= (0
,
2
,
6)
.
Method:
Find a vector
~
N
normal to two vectors
→
P
1
P
2
and
→
P
1
P
3
by computing
~
N
=
→
P
1
P
2
×
→
P
1
P
3
.
Use
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 Spring '08
 Auroux

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