MAT235, 20082009
Problem Set 2 – Solutions
Feel free to use calculators for your computations.
1.
(The numbers in the following problem are only approximately correct.) Let us think of the Earth as a
sphere of radius
6
,
340 km
, and let us put a system of
(
x,y,z
)
coordinates on it, with its axis of rotation along
the
z
axis and its centre at the origin. In this system of coordinates, the
Royal Observatory
in
Greenwich,
London
will be along the line of the vector
(1
,
0
,
1
.
255)
. The ancient city of
Timbuktu
in
Mali
is
33
◦
south of
London, and at its antipodal (that is, opposite) point on the sphere is the archipelago (and republic) of
Fiji
.
Fiji is located in the southwest Paciﬁc Ocean, and
30
◦
east of it is the island of
Tahiti
, the largest and capital
island of
French Polynesia
.
If a plane were to ﬂy from London to Tahiti, what is the shortest possible distance that it would need to
cover? Assume that the plane is ﬂying at a constant height of
10 km
.
Remarks.
1. The shortest route from a point
A
to a point
B
on a sphere is along the crosssection of the
sphere by the plane passing through
A
,
B
and the centre of the sphere.
2. Northsouth angles
always
express angles between the vectors joining the points with the centre of the
Earth.
3. Eastwest angles
always
express the angular displacement on a horizontal plane – that is, a plane which
is perpendicular to the Earth’s axis.
Solution:
By remark 3, the shortest possible distance for the plane is along the arc of a circle of radius
6
,
350 km
. We will compute the angle of this arc, in order to apply the formula: (Length of circular arc) =
(Radius) x (Angle of arc).
The vector
(1
,
0
,
1
.
255)
makes an angle of
Arctan(1
.
255) = 51
.
45
◦
with the positive
x
axis. Since Tim
buktu is
33
◦
south of London, it lies along the halfline of the vector
(1
,
0
,
tan(51
.
45
◦

33
◦
)) = (1
,
0
,
0
.
334)
.
Its antipodal point, Fiji, is therefore along the halfline of the vector
(

1
,
0
,

0
.
334)
. Going to the east cor
responds to counterclockwise rotation on the
x

y
plane, when viewed from the North Pole. Therefore
Tahiti, which is
30
◦
east of Fiji, will be along the halfline of the vector
(

1
·
cos 30
◦
,

1
·
sin 30
◦
,

0
.
334) =
(

√
3
2
,

1
2
,

0
.
334)
. Let
a
= (1
,
0
,
1
.
255)
,
b
= (

√
3
2
,

1
2
,

0
.
334)
. The angle
θ
between
a
and
b
satisﬁes:
cos
θ
=
a
·
b

a
 · 
b

.
We compute: