Graphics Reference
In-Depth Information
The direct approach is to solve the equation
n
•
p
= 0, that is, 2x + y - 3z = 0, for two
noncollinear points
v
1
and
v
2
. Alternatively, compute three noncollinear points
p
0
,
p
1
,
and
p
2
in
X
and set
v
1
=
p
0
p
1
and
v
2
=
p
0
p
2
. For example,
p
0
= (1,1,-1),
p
1
= (3,0,0),
and
p
2
= (0,6,0) would give
v
1
= (2,-1,1) and
v
2
= (-1,5,0). By construction these vectors
v
1
and
v
2
will also be a basis for
K
. The first approach that involves solving an equa-
tion for only two points rather than solving the equation 2x + y - 3z = 6 for three
points is obviously simpler; however, in other problems a plane may not be defined
by an equation.
Example 1.5.3 shows how one can find a basis for a plane if one knows some
points in it. A related question in the case of hyperplanes is to find the
equation
for
it given some points in it. To answer that question in
R
3
one can use the cross product.
Let
v
,
w
Œ
R
3
. Define the
cross product
v
¥
w
Œ
R
3
by
Definition.
(
)
v
¥=
vw
-
vw vw
,
-
vw vw
,
-
vw
.
(1.23)
23
32 31
13 12
21
Now, formula (1.23) is rather complicated. The standard trick to make it easier to
remember is to take the formal determinant of the following matrix:
i j k
vv v
ww w
Ê
ˆ
Á
Á
˜
˜
1
2
3
Ë
¯
1
2
3
The coefficients of the symbols
i
,
j
, and
k
will then be the x-, y-, and z-component,
respectively, of the cross product.
We shall look at the cross product and its properties more carefully later in Section
1.10. Right now we shall only make use of the fact that the cross product of two vectors
produces a vector that is orthogonal to both of these vectors, something easily checked
from the formula.
1.5.4. Example.
To find an equation for the hyperplane that contains the points
p
= (1,0,1),
q
= (1,2,0), and
r
= (0,0,3).
Solution.
We have that
(
)
(
)
¥=
(
)
pq
=
02 1
,,
-
,
pr
=-
102
,, ,
and
pq
pr
412
,, .
Therefore, an equation for the plane is
(
)
∑
(
(
)
-
(
)
)
=
412
,,
xyz
, ,
101
, ,
0
,
which reduces to
4
xy z
++ =.
2
6
If we compare arbitrary k-dimensional planes and hyperplanes, we see that the
former have so far only an explicit definition in terms of parameterizations whereas
