Game Development Reference
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4. (a) Right-handed.
(b) Swap y and z.
(c) Swap y and z.
5.
(a) Right-handed.
(b) x
us
← y
aero
,
y
us
←−z
aero
,
z
us
← x
aero
(c) x
aero
← z
us
,
y
aero
← x
us
,
z
aero
←−y
us
6. (a) CW
(b) CCW
(c) CCW
(d) CW
7. (a) 15
(b) 30
(c) 3840
(d) 2016840
(e) 5050
(a) π/6
(b) −π/4
(c) π/3
(d) π/2
(e) −π
8.
(f) 5π/4
(g) −3π/2
(h) 2.923
(i) 9.198
(j) −6π
(a) −30
o
(b) 120
o
(c) 270
o
(d) −240
o
(e) 360
o
9.
(f) 1
o
(g) 10
o
(h) −900
o
(i) 1800
o
(j) 36
o
10. The scarecrow should have said:
The sum of the squares of the legs of a right triangle is equal to the square of
the remaining side.
since the Pythagorean theorem is c
2
= a
2
+ b
2
, where a and b are the legs of the right
triangle and c is the hypotenuse.
(a) (sin(α)/ csc(α)) + (cos(α)/ sec(α)) = sin
2
(α) + cos
2
(α) = 1
(b) (sec
2
(θ)−1)/ sec
2
(θ) = 1−(1/ sec
2
(θ)) = 1−cos
2
(θ) = sin
2
(θ)
(c) 1 + cot
2
(t) = 1 + (cos
2
(t)/ sin
2
(t)) = (sin
2
(t)/ sin
2
(t)) + (cos
2
(t)/ sin
2
(t)) = (sin
2
(t) +
cos
2
(t))/ sin
2
(t) = 1/ sin
2
(t) = csc
2
(t)
(d) cos(φ)(tan(φ) + cot(φ)) = sin(φ) + (cos
2
(φ)/ sin(φ)) = (sin
2
(φ) + cos
2
(φ))/ sin(φ) =
1/ sin(φ) = csc(φ)
11.
B.2
Chapter 2
(Page 71.)
1.
(a) a is a 2D row vector. b is a 3D column vector. c is a 4D column vector.
(b) b
y
+ c
w
+ a
x
+ b
z
= 0 + 6 + (−3) + 5 = 8
2.
(a) “How much do you weigh?” Your weight is a scalar quantity. But the force of gravity,
which pulls you downwards, is a vector, and so if you said that weight was a vector
for that reason, you are also correct. (“My weight is 150 lbs of force in the downward
direction.”)
(b) “Do you have any idea how fast you were going?” The o
cer is probably referring
to the speed of your vehicle, which is a scalar quantity.
(c) “It's two blocks north of here.” Vector quantity.
(d) “We're cruising from Los Angeles to New York at 600 mph, at an altitude of 33,000 ft.”
The speed “600 mph” is a scalar quantity. Since New York is east of Los Angeles,
you could reasonably infer an eastward direction, so “600 mph eastward” is a veloc-
ity, which is a vector quantity. Likewise, “33,000 ft” is a scalar quantity, although
if you're a stickler, you might say that a direction of “up” is implied, in which case
“33,000 ft up” is a vector quantity.
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