A First Course in Linear Algebra, 2017a

560 Selected Exercise Answers
4.12.11 Wind: [ 0 50 ] . Direction it needs to travel: (3,5) √ 1
34
. Then you need 250 [ a b ] + [ 0 50 ]
to have this direction where [ a b ] is an appropriate unit vector. Thus you need
a 2 + b 2 = 1
250b + 50
250a
= 5 3
Thus a = 3 5 ,b = 4 5 . The velocity of the plane relative to the ground is [ 150 250 ] . The speed of the plane
relative to the ground is given by
√
(150) 2 +(250) 2 = 291.55 miles per hour
It has to go a distance of
√
(300) 2 +(500) 2 = 583.10 miles. Therefore, it takes
583.1
291.55 = 2 hours
4.12.12 Water: [ −2 0 ] Swimmer: [ 0 3 ] Speed relative to earth: [ −2 3 ] . It takes him 1/6 ofan
hour to get across. Therefore, he ends up travelling 1 6√ 4 + 9 =
1
6
√
13 miles. He ends up 1/3 mile down
stream.
4.12.13 Man: 3 [ a b ] Water: [ −2 0 ] Then you need 3a = 2andsoa = 2/3 and hence b = √ [ ]
5/3.
The vector is then .
2
3
√
5
3
In the second case, he could not do it. You would need to have a unit vector [ a b ] such that 2a = 3
which is not possible.
( )
4.12.17 proj ⃗ ⃗
D
F
= ⃗ F•⃗D
‖⃗D‖
(
⃗D
= ‖⃗D‖
4.12.18 40cos ( 20
180 π) 100 = 3758.8
4.12.19 20cos ( π
6
)
200 = 3464.1
4.12.20 20 ( cos π 4
)
300 = 4242.6
4.12.21 200 ( cos ( π
6
))
20 = 3464.1
4.12.22
⎡
⎣
−4
3
−4
⎤
⎡
⎦ • ⎣
0
1
0
⎦ × 10 = 30 You can consider the resultant of the two forces because of the properties
of the dot product.
⎤
( ⃗ D
‖⃗F‖cosθ) = ‖⃗D‖
)
‖⃗F‖cosθ
⃗u