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Therefore, the angle
α
n
between these vectors is
α
n
=
1

(
rs
)
n
1

rs
±
1

r
2
n
1

r
2
²

1
/
2
±
1

s
2
n
1

s
2
²

1
/
2
=
(1

r
2
)
1
/
2
(1

s
2
)
1
/
2
1

rs
±
1

(
rs
)
n
(1

r
2
n
)
1
/
2
(1

s
2
n
)
1
/
2
²
.
(c)
Since both

r

<
1 and

s

<
1, we have that lim
n
→∞
r
2
n
= 0, lim
n
→∞
s
2
n
= 0 and
also, lim
n
→∞
(
rs
)
n
= 0. Therefore
lim
n
→∞
`
n
=
1
√
1

r
2
and
lim
n
→∞
α
n
=
√
1

r
2
√
1

s
2
1

rs
.
You just computed the angle between two inﬁnite dimensional vectors! This will turn
out to be more than a mere curiosity.
4.7
Consider the vectors
a
=
h
a
b
i
. Find all vectors
h
c
d
i
that are orthogonal to
a
. That is ﬁnd conditions
on
c
and
d
in terms of
a
and
b
that are necessary and suﬃcient for this orthogonality.
SOLUTION
Computing the dot product,
³
a
b
´
·
³
c
d
´
=
ac
+
bd .
Thus, the vectors are orthogonal if and only if
ac
=
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.
 Fall '08
 Gladue
 Calculus, Vectors

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