Chapter 5: Hints and (partial) Solutions
Exercise 5.1
Start with distinct points
p
and
q
on the
n
sphere
S
n
. So
p
and
q
are unit vectors in
R
n
+1
. There
are three cases (well... only two really, the first is a special case of the third):
(a) If
p
⊥
q
(i.e.,
p
·
q
= 0) then the point
α
(
θ
) = (cos
θ
)
p
+ (sin
θ
)
q
for 0
≤
θ
≤
π/
2
,
(1)
belongs to
S
n
(why?, you can check that
k
α
(
θ
)
k
= 1 by computing
α
(
θ
)
·
α
(
θ
) and remembering that
p
·
q
= 0)
with
α
(0) =
p
and
α
(
π/
2) =
q
. Note that
α
will trace out a great circle on
S
n
if
θ
is allowed to range from 0
to 2
π
.
(b) If
q
=

p
, that is, if
p
and
q
are both on a line through the origin then intuitively
any
great circle containing
either
p
or
q
should contain the other (sketch a picture and you’ll see). Let’s check to make sure intuition is
correct here: first choose a point
e
∈
S
n
orthogonal to
p
(hence, orthogonal to
q
, ...could you construct such a
point? the next case will give you a clue on how to do this by first selecting any point
r
∈
S
n
not on the line
through
p
and
q
). Now proceed as in part (a) above, except with
e
in place of
q
: put
α
(
θ
) = (cos
θ
)
p
+ (sin
θ
)
e
and note that
α
(
θ
)
∈
S
n
for each
θ
∈
R
(you’ve already checked this in part (a)) and that
α
(0) =
p
and
α
(
π
) =

p
=
q
.
(c) If
p
and
q
are not on the same line then try this approach: replace
q
with another point
e
∈
S
n
such that
e
⊥
p
and
e
is in the same plane as the points
p
,
q
and
0
. Then, again as in (a) put
α
(
θ
) = (cos
θ
)
p
+ (sin
θ
)
e
. The
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 Spring '09
 Vector Space, Dot Product, Cos, ... ..., Rn+1

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