MAT 2355 Practice Problems.


cos ✓
sin ✓
x0
1. Let ✓ 2 R, set R✓ =
and fix v0 =
sin ✓ cos ✓
y0
2
2
f : R ! R by f (v) = R✓ v + v0
2 R2 , and define
a) Show that f is an isometry.
b) Show that if v0 = 0, then f (u) · f (v) = u · v for every u, v 2 R2 .
2. [Exercise 1.2.1] a) Show that every isometry f : Rn ! Rn is a one-to-one function. (i.e show that if P 6= Q are 2 di↵erent points in Rn , and f is an isometry,
then f (P ) 6= f (Q).)
b) Assume that an isometry f : Rn ! Rn has an inverse f 1 : Rn ! Rn (i.e.
f (f 1 (P )) = P and f 1 (f (Q)) = Q for all P, Q 2 Rn . Show that f 1 must also
be an isometry.
3. [Exercise 1.2.2] Fix P 2 E2 and r > 0. Let C = {Q 2 E2 | d(P, Q) = r} be the
circle with centre P and radius r. If f : E2 ! E2 is an isometry, prove that f (C)
is the circle with centre f (P ) and radius r. (i.e. show that isometries of E2 map
circles to circles).
4. a) Suppose R2 3 (a, b) 6= 0 and c 2 R. Let be a line in R2 . Find a formula for
the reflection RL : R2 ! R2 in the line
L = {(x, y) 2 R2 | ax + by + c = 0}.
b) Check that your formula in (a) really does define an isometry.
2
c) Check that your formula in (a) satisfies RL
(P ) = RL (RL (P )) = P for all P 2 R2 .
5. Show that if f : Rn ! Rn is an isometry and P and Q are points in Rn , then
f (P Q) = f (P )f (Q), i.e. that f maps the segment with endpoints P and Q to the
segment with endpoints f (P ) and f (Q).
2
0
6. Let A = 4 1
0
a) Show that
3
0 1
0 0 5 and consider the map f : R3 ! R3 defined by f (x) = Ax.
1 0
f is an isometry.
2
b) Is f a reflection ? If so, find the plane of reflection, and if not, give reasons.
7. Suppose n 6= 0 is a vector in Rn and let H = {v 2 Rn | v · n = 0} be the
hyperplane through the origin with normal vector n. If RH : Rn ! Rn is the
reflection in H, show that the matrix A of RH is a symmetric matrix. (i.e. satisfies
At = A).
8. (a) Find 2 lines L0 and L1 so that reflection in L0 followed by reflection in L1
is the isometry f : R2 ! R2 defined by f (x, y) = (x + 1, y 2). Check that your
reflections achieve the desired result.
(b) Find 2 planes H0 and H1 in R3 so that reflection in H0 followed by reflection
in H1 is the isometry f : R3 ! R3 defined by f (x, y, z) = (x + 1, y + 2, z 2).
Check that your reflections achieve the desired result.
9. Let O(n) = {A | A is an n ⇥ n matrix with AAt = In } be the set of n by n
orthogonal matrices. Show that
a) In 2 O(n)
b) If A, B 2 O(n), then AB 2 O(n) and that
c) If A 2 O(n), then A 1 2 O(n).
(i.e. Under the operation of matrix multiplication, O(n) is a group.)
10. Show that if A 2 O(n) and b 2 Rn are fixed, then f : Rn ! Rn is defined by
f (x) = Ax + b is indeed an isometry.
11. a) Show that the composition of two isometries is an isometry.
b) Show that any isometry f : Rn ! Rn has an inverse, that is, that there is a
function g : Rn ! Rn such that f (g(P )) = P and g(f (P )) = P for all P 2 Rn .
c) Show that the inverse g of the isometry f is an isometry.

cos ✓
sin ✓
a line L through the origin, and the line of reflection.
12. Show that f : R2 ! R2 defined by f (v) =
sin ✓
v is a reflection in
cos ✓
13. Let L1 and L2 be two lines in R2 with direction vectors {v1 , v2 }, which meet at
a single point P . Show that RL2 RL1 is the rotation about P , in the positive sense
3
determined the orientation {v1 , v2 }, through the angle which is twice the (acute)
angle between L1 and L2 .
14. (Bonus) Prove or find a counterexample: if any isometry f : R3 ! R3 satisfies
f (0) = 0, then f (u) ⇥ f (v) = f (u ⇥ v) or f (u) ⇥ f (v) = f (u ⇥ v)for all u, v 2 R3 ,
where “⇥” denotes the cross product in R3 . If it is true, what property of f
determines the choice of sign?
15. (a) Let P 6= Q be distinct points in Rn . Show that H = {v 2 Rn | ||v
||v Q||} is a hyperplane and that reflection in H interchanges P and Q.
P || =
(b) Suppose a 2 Rn , b 2 R with ||a|| = 1 and let H = {v 2 Rn | v · a = b} be a
hyperplane in Rn . Show that if P and Q are any points in Rn such that P Q
is parallel to a , and P · a b = (Q · a b) > 0, then H = {v 2 Rn | ||v P || =
||v Q||}. (What does “P · a b = (Q · a b)” mean geometrically?)
16. Let T1 be the triangle whose vertices are (1, 1),p(1, 4)
and (4, 4) and T2 the
p
p
triangle whose vertices are (0, 0), ( 3 2, 0) and ( 3 2 2 , 3 2 2 ). Find an isometry f
of R2 such that f (T1 ) = T2 .
17. If H and K are perpendicular planes in R3 , show that RH (K) = K, where RH
denotes the reflection in H.
18. Show that the reflection RH in a hyperplane H ⇢ Rn reverses orientation.
19. Consider the plane H = {v 2 R3 | v · a = 0} which contains the origin and
which is perpendicular to the unit vector a 2 R3 . Suppose ✓ 2 [0, 2⇡], and let
⇢ : R3 ! R3 be the isometry which is rotation by ✓ about the line through the
origin with direction a. As usual, let RH denote the reflection in H. Show that
⇢RH = RH ⇢.
20. Suppose 3 distinct points A, B, C lie on a line L, with B between A and C, and
let D be a point not on L.
(a) Express the fact that “B is between A and C” algebraically.
4
(b) Using (a) or otherwise, show that if B is between A and C and f is an isometry,
then f (B) is between f (A) and f (C).
(c) Show that \ABC = ⇡ and that \DBA + \DBC = ⇡. (Hint: use a suitable
isometry, and (b), to simplify your life.)
21. Suppose A, B, C, D are points in the plane with A 6= D, and let L be the line
L through A and D. Suppose that B and C are on the same side of L.
(a) Express the fact that “B and C are on the same side of the line through A and
D” algebraically (Hint: use orientations in the plane).
(b) Using (a) or otherwise, show that if B and C are on the same side of the line
through A and D and f is an isometry, then f (B) and f (C) are on the same side
of the line through f (A) and f (D)
(c) If \CAD = \BAD, show that A, B and C are collinear. (Hint: use a suitable
isometry, and (b), to simplify your life.)
22. (Bonus) Suppose A is a real, n ⇥ n orthogonal matrix.
a) Show that if is a (possibly) complex eigenvalue of A, then | | = 1, where | |
denotes the modulus of the complex number .
b) Show that if is an eigenvalue of A then ¯ (where the bar denotes complex
conjugation) is also an eigenvalue of A.
c) Show that eigenvectors of A corresponding to di↵erent eigenvalues are orthogonal.
d) Suppose v 2 Cn is a (possibly complex) eigenvector of A with eigenvalue .
Show that v̄ is an eigenvector with eigenvalue ¯ .
e) Show that if v is a eigenvector of A with eigenvalue = ei✓ 2
/ R, and we
v+v̄
v v̄
define v1 = 2 , and v2 = 2i , show that {v1 , v2 } is an orthogonal subset of
Rn , with ||v1 || = ||v2 ||, and finally that
Av1 = (cos ✓)v1 + (sin ✓)v2 ,
(*)
Av2 =
(sin ✓)v1 + (cos ✓)v2 .
5
23. (Bonus) Suppose A is a real, 3 ⇥ 3 orthogonal matrix with det A = 1. Show
that there is an orthonormal basis {v1 , v2 , v3 } of R3 and a real number ✓ 2 [0, 2⇡)
such that
Av1 = (cos ✓)v1 + (sin ✓)v2 ,
Av2 =
(**)
That is, if P = [ v1
(sin ✓)v1 + (cos ✓)v2 ,
Av3 = v3 .
v2
v3 ], then
2
cos ✓
1
P AP = 4 sin ✓
0
sin ✓
cos ✓
0
3
0
05
1
24. (Bonus) Suppose A is a real, 3 ⇥ 3 orthogonal matrix with det A = 1. Show
that there is an orthonormal basis {v1 , v2 , v3 } of R3 and a real number ✓ 2 [0, 2⇡)
such that either
Av1 = (cos ✓)v1 + (sin ✓)v2 ,
(i)
(that is, if P = [ v1
Av2 = (sin ✓)v1 + (cos ✓)v2 ,
Av3 = v3 ,
v2
cos ✓
1
v3 ], then P AP = 4 sin ✓
0
sin ✓
cos ✓
0
Av1 = (cos ✓)v1 + (sin ✓)v2 ,
(ii)
(that is, if P = [ v1
2
Av2 = (sin ✓)v1
Av3 = v3 .
v2
(cos ✓)v2 ,
v3 ], then
2
cos ✓
1
P AP = 4 sin ✓
0
sin ✓
cos ✓
0
3
0
0 5 .)
1
3
0
0 5) , or
1
6
25. (Bonus) Suppose A is a real, 4 ⇥ 4 orthogonal matrix with det A = 1. Show
that there is an orthonormal basis {v1 , v2 , w1 , w2 } of R4 and two real numbers
✓, ' 2 [0, 2⇡) so that
Av1 = (cos ✓)v1 + (sin ✓)v2 ,
Av2 =
(sin ✓)v1 + (cos ✓)v2 ,
Aw1 = (cos ')w1 + (sin ')w2 ,
(*)
That is, if P = [ v1
Aw2 =
v2
w1
(sin ')w1 + (cos ')w2 .
w2 ], then
2
cos ✓
6 sin ✓
P 1 AP = 4
0
0
sin ✓
cos ✓
0
0
0
0
cos '
sin '
3
0
0 7
5
sin '
cos '
26. Show that any two line segments AB and CD with non-zero lengths in R3 are
similar.
2
0
2
2
27. Let g : S ! S be defined by g(v) = Av where A = 4 1
0
0
0
1
3
1
0 5.
0
a) Show that g is an orientation reversing isometry of S2 .
b) Find a great circle C, a point a 2 S2 and ✓ 2 [0, 2⇡) such that g is rotation
by ✓ about a followed by reflection in C.
c) Check your answer in (b) by an explicit computation.
28. Suppose ⇢ and ⇢0 are rotations of S2 such that ⇢⇢0 = ⇢0 ⇢. Show that either
i) ⇢ and ⇢0 are rotations about the same point, by any 2 angles, or,
ii) ⇢ and ⇢0 are rotations, both by ⇡, about “perpendicular points”.
29. Show that any two line segments AB and CD with non-zero lengths in R3 are
similar.
7
Recall from class that if
of is defined as
: [0, 1] ! R3 is a smooth curve, then the length l( )
l( ) :=
Z
0
1
|| 0(t)||dt.
Recall as well that if P, Q 2 S2 , the spherical distance d(P, Q) from P to Q is
the length of the shortest (smooth) path on S2 which starts at P and ends at Q.
30. (Bonus) Suppose that f : R3 ! R3 is a Euclidean isometry, and that
[0, 1] ! R3 is a smooth curve.
i) Show that l( ) = l(f
:
).
Now suppose in addition that f (0) = 0.
ii) Show that f (S2 ) = S2 .
iii) Conclude that if P, Q 2 S2 , then d(P, Q) = d(f (P ), f (Q)). (You may not assume
in this question that d(P, Q) = arccos(P · Q). This will be established in the next
question, which will rely on this question!)
31. Suppose P, Q 2 S2 . Recall from class that if that if either P or Q is N , then
d(P, Q) = arccos(P · Q). Now suppose neither P nor Q is N , and let H be the plane
equidistant from P and N .
i) Show that the reflection RH in H fixes the origin, that and so conclude that
H \ S2 is a great circle.
ii) Show that d(P, Q) = d(N, RH Q).
iii) Conclude that d(P, Q) = arccos(P · Q).
32. Given any 2 points P, Q 2 S2 with P 6= Q show that there is a great circle
containing P and Q. Show that this great circle is unique unless P 2 {Q, Q}, and
that in these cases there are infinitely many great circles containing P and Q.
33. A pair of points of the form {P, P } is called antipodal.
a) Show that any two distinct great circles intersect in a pair of antipodal points.
b) Show that antipodal points remain antipodal after an isometry of S2 .
2
3
0
0
1
34. Let g : S2 ! S2 be defined by g(v) = Av where A = 4 1 0
0 5.
0
1 0
8
a) Show that g is an orientation reversing isometry of S2 .
b) Find a great circle C, a point a 2 S2 and ✓ 2 [0, 2⇡) such that g is rotation
by ✓ about a followed by reflection in C.
c) Check your answer in (b) by an explicit computation.
35. Let a 2 S2 , 0 6= b 2 R and H = {v 2 R3 | a · v = b}. Show that there is a
non-zero point h0 2 H such that H = {v 2 R3 | h0 · v = ||h0 ||2 }. Show that h0 is
the closest point on H to the origin.
36. A subset C ⇢ R3 is a circle if there is a plane K, a point k 2 K and 0 < r 2 R
such that C = K \ {v 2 R3 | ||v k|| = r}.
a) Show that the intersection of any plane K with S2 is either empty, consists of
a single point, or is a circle in R3 . (Hint: use the previous question.)
b) Show that if C = K \ {v 2 R3 | ||v k|| = r} is a circle, then for any normal
a to K,
C = K \ {v 2 R3 | ||v (k + a)||2 = r2 + ||a||2 }.
37. Let K be a plane, k 2 K and r > 0 be given, and suppose the circle C =
K \ {v 2 R3 | ||v k|| = r} lies in S2 . Show that C = K \ S2 .
2
0
2
2
38. Let g : S ! S be defined by g(v) = Av where A = 4 1
0
0
0
1
a) Show that g is an orientation reversing isometry of S2 .
3
1
0 5.
0
b) Find a great circle C, a point a 2 S2 and ✓ 2 [0, 2⇡) such that g is rotation
by ✓ about a followed by reflection in C.
c) Check your answer in (b) by an explicit computation.
2
0 0
2
2
39. Let f : S ! S be defined by f (v) = Bv where B = 4 1 0
0 1
a) Show that f is an orientation preserving isometry of S2 .
3
1
0 5.
0
b) Find a point b 2 S2 and ' 2 [0, 2⇡) such that f is rotation by ' about b.
9
c) Check your answer in (b) by an explicit computation.
40. Suppose A 6= B 2 S2 , and let : [0, 1] ! S2 be the geodesic path from A to
B. Show that if u = 0 (0), then u · B
0. (Hint: first reduce to the case where
A = N = (1, 0, 0), and then use the formula we found in class for .)
Use only the cosine rule for the next two questions. (Final answers to 4 significant
figures only.)
41. Solve the spherical triangle ABC if
a) A = 75 , b = 80 , and c = 85 .
b) a = 75 , b = 80 , and c = 85 .
42. A plane flies from Ottawa (45 240 N 75 430 W ) along a great circle to Bejing
(39 550 N 116 250 E).
a) Find its heading as it leaves Ottawa and its heading as it arrives in Bejing.
b) If the radius of the earth is 6378km, and the plane averages 1000km per hour,
how long will the trip take? (to the nearest minute.)
c) What is the distance of closest approach to the north pole during the flight?
(to the nearest kilometre.)
43. Let H be the plane {(x, y, z) | z = 1} and suppose f : S2 = {(x, y, z) 2 S2 |
z < 0} ! H Prove that central projection does not always preserve angles.
44. Show that the images under the stereographic projection of great circles on S2
through N = (0, 0, 1) are lines, and describe these lines.