Answers to the Exercises - 3D Math Primer for Graphics and Game Development - page 784

The sums can be recognized as the Taylor series expansion for cosine and sine; therefore,

e ix = cos x + i sin x.

This equation is known as Euler's formula. Substituting x = π and moving everything to
the left hand side gives us Euler's identity, a beautiful equation that links together five
important mathematical constants:

e iπ + 1 = 0.

12. This one has a few tricks. First, we need to compute the actual radius of the orbit, taking
into account Earth's (average) radius of 6,371 km, as

r = 6, 371 km + 340 km = 6, 711 km.

Now the length of the circular orbit is just the circumference of a circle with this radius,
which can be computed using elementary geometry:

C = 2πr = 2π(6, 711 km) = 4.217×10 4 km.

Finally, we divide this distance by the average speed to get the orbital period:

P = C/s = (4.217×10 4 km)/(27, 740 km/hr) = 1.520 hr = 91.21 min.

The centripetal acceleration can be computed by Equation (11.29):

2 2

27, 740 km
hr

× 1 hr
3, 600 s

7.706 km
s

a = s 2
r

= 0.008849 km
s 2

m
s 2 .

= = = 8.849

6, 711 km 6, 711 km

B.12 Chapter 12

(Page 640.)

1. We must consider all the forces acting on the fan, the air, and the boat. As the fan rotates,
a force exists between the fan and the air, which wants to push the air forward and the
fan backwards. Since the fan does not accelerate backwards, we know that there must
be some force opposing it, and this force comes from the force of friction provided by the
boat. But then this means the boat is receiving a backwards force, and this backwards
force counteracts any force eventually received by the wind hitting the sail.
2. First, we identify four bodies: the girl, the boy, the rope, and Earth. Next, we identify the
active tension and friction forces:

T g,r Girl pulls on rope T r,g Rope pulls on girl

F g,e Girl pushes on Earth F e,g Earth pushes on girl

T b,r Boy pulls on rope T r,b Rope pulls on boy

F b,e Boy pushes on Earth F e,b Earth pushes on boy