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