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Problem Set #5
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Section:
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Problem #1
.
Let
I
and
J
denote any two orthogonal unit vectors in
R
3
which both happen to have a
z
–com
ponent of zero.
Find, to within a sign, the value of
I
×
J
– and show all of your reasoning.
Math 32A
Lecture #5
Fall 2007
C
REDIT
2
1
0
Problem #2
.
Let
A
(
t
) and
B
(
t
) denote two vector valued functions of time.
Derive a formula for the deriva
tive of the cross product:
d
dt
A
×
B
.
Hint: You can try a derivation on the (primitive) de±
nition which involves the basis vectors.
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Problem #4
.
Let
r
A
and
r
B
denote (non–zero) vectors in
R
n
.
Although these have a dot product, you may
not
assume anything like the formula involving dot products and cosines of angles.
In fact, something like this
is what we will show in the form of an inequality.
Part I.
Show that for any scalar
λ
,

r
A

2
+
λ
2

r
B

2
≥
2
λ
r
A
•
r
B
.
Part II.
Find the optimal constant
λ
which makes the left and right side of the above inequality as close as pos
sible.
Plugging in
that
value of
λ
, obtain a famous inequality involving 
r
A
, 
r
B
 and
r
A
•
r
B
.
C
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 Fall '08
 GANGliu
 Vectors, Dot Product

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