Section 6.1b
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Transcript of Section 6.1b
Section 6.1bSection 6.1b
Direction AnglesDirection Angles
Velocity, SpeedVelocity, Speed
Let’s start with a brain exercise…Find the unit vector in the direction of the given vector. Writeyour answer in (a) component form and (b) as a linearcombination of the standard unit vectors i and j.
u 3, 4
Unit Vector:u
u 22
3, 4
3 4
3, 4
5
3 4
,5 5
With standard unit vectors:3 4i j5 5
Direction Angle – the angle O that a vector makes with the positive x-axis
x
yv
θθ
θ
|v|cos
|v|sin using trigonometry…using trigonometry…
Thus,
v = (|v|cos 0)i + (|v|sin 0)j
And the unit vector in the direction of v is
u = = (cos 0)i + (sin 0)jv
|v|
Guided PracticeFind the components of the vector v with direction angle123 and magnitude 5.
v 5cos123 ,5sin123
2.723,4.193 Does this answer make sense graphically ???
Guided PracticeFind the magnitude and direction angle of each vector.
w 13 θ 33.690
w = 3, 2
Guided PracticeFind the magnitude and direction angle of each vector.
w 89 θ 302.005
w = 5i – 8j
Guided PracticeFind the vector v with the given magnitude and the samedirection as u.
u = –5, 7v = 5 Can we see thisproblem in a graph?
First, find the unit vector in the direction of u:
2 2
5,7u
u 5 7
5,7
74
5 7
,74 74
Now, simply multiply this vector by |v| (the magnitude of v):
5 7v v ,
74 74
25 35,
74 74
2.906,4.069
Velocity – distance covered per unit time – this is a vectorb/c it has both magnitude and direction
Speed – the magnitude of velocity (a scalar)
Ex: An aircraft is flying on a bearing of 65 at 500mph. Find the component form of the velocity of the plane
v 500cos 25 ,500sin 25 453.154,211.309
Start with a graph…do you remember the definition of bearing ?
Ex: An aircraft is flying on a compass heading (bearing) of 350 at 355 mph. A wind is blowing with the bearing 285 at 42 mph. Find (a) the component form of the aircraft’s velocity, and (b) the actual ground speed and direction of the aircraft.
v 102.214,360.477
Actual speed = 374.688 mphDirection = 344.169 bearing
(a)
(b)
Cool problem…Three forces with magnitudes 100, 50, and 80 lb, act on anobject at angles of 50 , 160 , and –20 , respectively. Find thedirection and magnitude of the resultant force.
F1 F2 F3
Start with a diagram:F1
F2
F3
100 lb
80 lb
50 lb
160 50
–20
More of our cool problem…Three forces with magnitudes 100, 50, and 80 lb, act on anobject at angles of 50 , 160 , and –20 , respectively. Find thedirection and magnitude of the resultant force.
F1 F2 F3
Find the component form of each force:
1F 100cos50 ,100sin 50
2F 50cos160 ,50sin160
3F 80cos 20 ,80sin 20
R 1 2 3F F F F 92.470,66.344 Sum the forces:
Still more for our cool problem…Three forces with magnitudes 100, 50, and 80 lb, act on anobject at angles of 50 , 160 , and –20 , respectively. Find thedirection and magnitude of the resultant force.
F1 F2 F3
Magnitude of the resultant force:
2 292.470 66.344RF 113.808 lb
FR
113.808 lb
92.470 lb
66.344 lb
Direction of the resultant force:
66.344tan
92.470
1 66.344tan92.470
35.658
More fun examples!!!More fun examples!!!A pilot’s flight plan has her flying due east from Flagstaff.There is a 65-mph wind bearing 60 , and the aircraft hasa 450 mph speed with no wind. What heading shouldthe pilot follow, and what will be the aircraft’s resultantground speed?
Heading = 94.142 , Speed = 505.116 mphHeading = 94.142 , Speed = 505.116 mph