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Section 6.1b

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Section 6.1b

Direction Angles

Velocity, Speed

Let’s start with a brain exercise…

Find the unit vector in the direction of the given vector. Write

your answer in (a) component form and (b) as a linear

combination of the standard unit vectors i and j.

Unit Vector:

With standard unit vectors:

Direction Angle – the angle O that a vector makes with

the positive x-axis

y

v

|v|sin

using trigonometry…

x

|v|cos

Thus,

v = (|v|cos 0)i + (|v|sin 0)j

And the unit vector in the direction of v is

v

u = = (cos 0)i + (sin 0)j

|v|

Guided Practice

Find the components of the vector v with direction angle

123 and magnitude 5.

Does this answer make sense graphically ???

Guided Practice

Find the magnitude and direction angle of each vector.

w = 3, 2

Guided Practice

Find the magnitude and direction angle of each vector.

w = 5i – 8j

Guided Practice

Find the vector v with the given magnitude and the same

direction as u.

Can we see this

problem in a graph?

v = 5

u = –5, 7

First, find the unit vector in the direction of u:

Now, simply multiply this vector by |v| (the magnitude of v):

Velocity – distance covered per unit time – this is a vector

b/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

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.

(a)

(b)

Actual speed = 374.688 mph

Direction = 344.169 bearing

Cool problem…

F

F

F

2

3

1

Three forces with magnitudes 100, 50, and 80 lb, act on an

object at angles of 50 , 160 , and –20 , respectively. Find the

direction and magnitude of the resultant force.

Start with a diagram:

F

1

100 lb

F

2

160

50

50 lb

–20

80 lb

F

3

More of our cool problem…

F

F

F

2

3

1

Three forces with magnitudes 100, 50, and 80 lb, act on an

object at angles of 50 , 160 , and –20 , respectively. Find the

direction and magnitude of the resultant force.

Find the component form of each force:

Sum the forces:

Still more for our cool problem…

F

F

F

2

3

1

Three forces with magnitudes 100, 50, and 80 lb, act on an

object at angles of 50 , 160 , and –20 , respectively. Find the

direction and magnitude of the resultant force.

Magnitude of the resultant force:

lb

Direction of the resultant force:

F

R

113.808 lb

66.344 lb

92.470 lb

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 has

a 450 mph speed with no wind. What heading should

the pilot follow, and what will be the aircraft’s resultant

ground speed?

Heading = 94.142 , Speed = 505.116 mph