Graphics Reference
In-Depth Information
Section 1.9
1.9.1.
Let
12 3
25 4
348
-
-
Ê
ˆ
Á
˜
A =
.
Ë
--
¯
Find a nonsingular matrix C so that CAC
T
is a diagonal matrix.
Section 1.10
1.10.1.
Prove Proposition 1.10.3. (Hint: First show that, if
e
1
,
e
2
, and
e
3
are the standard basis
vectors in
R
3
, then
e
1
¥
e
2
=
e
3
,
e
1
¥
e
3
=-
e
2
, and
e
2
¥
e
3
=
e
1
.)
1.10.2.
Prove Proposition 1.10.4.
Note:
The properties will not be hard to prove if one uses the definition and basic
properties of determinants. This shows once again how valuable a good definition is
because some textbooks, especially in the physical sciences, deal with cross products
in very messy ways. Although it is our intuition which leads us to useful concepts, it
is usually a good idea not to stop with the initial insight but probe a little further and
really capture their essence.
Prove that if
u
,
v
Œ
R
3
are orthogonal unit vectors, then (
u
¥
v
) ¥
u
=
u
.
1.10.3.
Let
u
,
v
,
w
Œ
R
3
. Prove
1.10.4.
u
v
w
Ê
ˆ
Á
Á
˜
(a)
u
•(
v
¥
w
) =
det
˜
.
Ë
¯
(b)
u
•(
v
¥
w
) =
v
•(
w
¥
u
) =
w
•(
u
¥
v
).
(The quantity
u
•(
v
¥
w
) is called the
triple product
of
u
,
v
, and
w
.)
