Graphics Reference
In-Depth Information
It follows that
2
5
1
30
1
6
1
2
1
2
Ê
ˆ
Ê
ˆ
-
0
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
5
30
1
6
1
6
2
6
1
6
(
)
=
(
)
=
(
)
(
)
xyz
¢¢¢
, ,
Mxyz
,,
xyz
,,
-
1
0
-
+-
010
, ,
1
5
2
30
2
6
1
3
1
3
1
3
-
-
-
-
Ë
¯
Ë
¯
For emphasis, we note that the solutions to the last four examples are not unique.
For example, to define M in Example 2.5.2.4 we could have picked
any
orthonormal
basis (
u
1
,
u
2
) for
X
. It is essential however that one picks an
orthonormal
basis,
namely, a basis that consists of
unit
vectors that are
mutually orthogonal
. If either
of these conditions does not hold, then answers will be wrong.
2.6
E
XERCISES
Section 2.2
2.2.1.
Prove Theorem 2.2.1(2) and (3).
2.2.2.
Prove that motions send triangles to triangles.
2.2.3.
Prove that motions send rays to rays.
Section 2.2.1
2.2.1.1.
Prove Theorem 2.2.1.1.
2.2.1.2.
Prove Proposition 2.2.1.2(2).
Section 2.2.2
2.2.2.1.
Prove Proposition 2.2.2.7(1).
2.2.2.2.
Find the rotation about the point (1,2) through an angle of p/6.
2.2.2.3.
Let R be a rotation about the origin through an angle p/3. Let
L
be the line determined
by the two points (2,4) and (4,4-2/÷
-
). Show by direct computation that the angles
that
L
and
L
¢=R(
L
) make with the x-axis differ by p/3.
Find the rotation R about the point (2,3) that sends (6,3) to (4,3 +÷
-
).
2.2.2.4.
2.2.2.5.
Let R be the rotation about (-1,2) through an angle of -p/6. Let
L
be the line deter-
mined by the points (2,4) and (5,1). Find the equation for
L
¢=R(
L
).
2.2.2.6.
If R is the rotation about the origin through an angle of p/3 degrees and if T is the
translation with translation vector (-1,2), then find the equation for RT and describe
the map in geometric terms as precisely as possible.
