Graphics Reference
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x
=+
=-
=
23
t
yt
z
7
and orthogonal to the plane x - z = 2 .
Solution. If n = (a,b,c) is a normal for X , then n must be orthogonal to the direc-
tion vector (3,-1,0) for the given line and orthogonal to the normal (1,0,-1) for the
given plane, that is,
3
a-=
0
and
a-=0.
Solving these two equations gives that b = 3a and c = a. In other words, (a,3a,a) is a
normal vector for X . It follows that
(
) (
(
) - (
)
) =
131
,,
xyz
, ,
132
,,
0
or
xyz
++=
3
12
is an equation for X .
We finish this section with two more definitions. The first generalizes the half-
planes R + and R - .
Let p 0 , n Œ R n with n π 0 . The sets
Definition.
{
}
n
(
)
pR n pp
Œ
-
0
0
and
{
}
n
(
) £
pR n pp
Œ
-
0
0
are called the halfplanes determined by the hyperplane n • ( p - p 0 ) = 0. A halfline is
a halfplane in R .
A hyperplane in R n divides R n into three parts: itself and the two halfplanes on
either “side” of it. Figure 1.7 shows the two halfplanes in the plane defined by the line
(hyperplane) 2x + 3y - 6 = 0.
Sometimes one needs to talk about the smallest plane spanned by a set.
Let X Õ R n . The affine hull or affine closure of X , denoted by aff ( X ), is
Definition.
defined by
() {
}
aff XP P
is a plane which contains
X
.
The following lemma justifies the definition of the affine hull of a set:
 
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