Graphics Reference
In-Depth Information
1.5.16.
Find the point-normals equation for the line
x
=-
=
=+
13
2
3
t
yt
z
t
1.5.17.
Determine whether the halfplanes 2y - x ≥ 0, y - 2x + 2 ≥ 0, and -4y + 2x + 4 £ 0 have
a nonempty intersection or not.
Let
V
and
W
be subspaces of
R
n
of dimension s and t, respectively. Assume that s + t
≥ n. Prove that
V
is transverse to
W
if and only if one of the following holds:
(a)
1.5.18.
V
+
W
=
R
n
.
(b)
If
v
1
,
v
2
,...,
v
s
and
w
1
,
w
2
,...,
w
t
are bases for
V
and
W
, respectively, then the
vectors
v
1
,
v
2
,...,
v
s
,
w
1
,
w
2
,...,
w
t
span
R
n
.
Definition.
Any subset
X
in
C
n
of the form
1.5.19.
{
}
Xp v v
=+ +
t
t
++
...
t
v
t
,
t
,...,
t
Œ
C
,
11
2 2
kk
1 2
k
where
p
is a fixed point and the
v
1
,
v
2
,...,
v
k
are fixed linearly independent vectors in
C
n
is called a
complex k-dimensional plane
(through
p
). If k = 1, then
X
is called a
complex line
.
Prove that a complex line in
C
2
can also be expressed as the set of points (x,y) Œ
C
2
satisfying an equation of the form
(a)
ax
+=,
by
c
for fixed a, b, c Œ
C
with (a,b) π (0,0).
Prove that the real points of a complex plane in
C
n
lie on a plane in
R
n
.
(b)
Section 1.6
1.6.1.
Prove Lemma 1.6.1.
Determine whether the following pairs of ordered bases of
R
2
1.6.2.
determine the same
orientation:
(a) ((1,-2), (-3,2)) and ((1,0), (-2,3))
(b) ((-1,1), (1,2)) and ((1,-2), (1,-4))
Solve this exercise in two ways: First, use only the definition of orientation and then
check your answer using the matrix approach of Lemma 1.6.4.
1.6.3.
Why is “Does ((1,-2), (-2,4)) induce the standard orientation of the plane?” a mean-
ingless question?
1.6.4.
(a)
Find a vector (a,b) so that the basis ((-2,-3), (a,b)) determines the standard orien-
tation of the plane.
(b)
Find a vector (a,b,c) so that the basis ((2,-1,0), (-2,-1,0), (a,b,c)) determines the
standard orientation of 3-space.