VECTORSVECTORS

Scalars are quantities which are fully described by a magnitude alone.

Vectors are quantities which are fully described by both a magnitude and a direction.

Do you remember the difference between a scalar and a vector?

1. The length of the line represents the magnitude and the arrow indicates the direction.2. The magnitude and direction of the vector is

clearly labeled.

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn precisely to length in accordance with a chosen scale.

Scaling!!!Scaling!!!

Direction!!!Direction!!!

Sometimes vectors will be directed due East or due North. However we will encounter vectors in all sorts of directions and be forced to find the angle!

Compass Coordinate System

Compass Coordinate System

• Δx = 30 m @ 20º E of N

• V = 20 m/s @ 30º W of N

• a = 10 m/s2@ 40º W of S

• F = 50 N @ 10º S of E

Navigational System?

Use the same scale for all vector magnitudes

E

S

N

W

To Draw direction:

Ex. 20º E of N: Start w/ North and go 20° East

All these planes have the same reading on their speedometer. (plane speed not speed with respect to the ground (actual speed)

What factor is affecting their velocity?

A B C

Easy Adding…

The resultant is the vector sum of two or more vectors.

1. Select an appropriate scale (e.g., 1 cm = 5 km)

2. Draw and label 1st vector to scale.

*The tail of each consecutive vector begins at the head of the most recent vector*

3. Draw and label 2nd vector to scale starting at the head of the 1st vector.

4. Draw the resultant vector (the summative result of the addition of the given vectors) by connecting the tail of the 1st vector to the head of the 2nd vector. (initial to final pt.)

5. Determine the magnitude and direction of the resultant vector by using a protractor, ruler, and the indicated scale; then label the resultant vector.

A. Tailwind

(with the wind)

120 km/h20 km/h

80 km/h

100 km/h

100 km/h

20km/h

=

B. Headwind

(against the wind)

=

80 km/h

100 km/h

60 km/h

Resultant

C. 90º crosswindUsing a ruler and your scale, you can determine the magnitude of the resultant vector. Or you could use the Pythagorean Theorem.

Then using a protractor, you can measure the direction of the resultant vector. Or you could use trigonometry to solve for the angle.

1. Find the resultant force vector of the two forces below.

25 N due East, 45 N due South

25 N, East

45 N, South

Decide ona scale!!!

51 N59º S of E

51 N31º E of S

0o

N

180o

270o

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

E

N

W

S

An airplane is flying 200mph at 50o N of E. Wind velocityis 50 mph due S. What is the velocity of the plane?

Scale: 50 mph = 1 inch

200 mph

50 mph

VR = 165 mph @ 40° N of E

2. Find the resultant velocity vector of the two velocity vectors below.

700 m/s @35 degrees E of N; 1000 m/s @ 30 degrees N of W

V1

Vr

V2

A bear walks one mile south, then one

mile west, and finally walks one

mile north. After his brisk walk, the bear ends back where he

started.

What color is the bear???

Intro to Vectors Warm-up

In what direction is the leash pulling on the dog?

What would happen to the upward and rightward Forces if the Force on the chain were smaller?

1) Find the resultant Magnitude:__________

of the two vectors Direction:___________

Vector #1 = 20.5 N West

Vector #2 = 14.3 N North

Vector Diagram

24.99 N

34.90º N of W

V2

V1

VR=?

2) Find the component of the

resultant = 255m 27º South of East

Vector # 1 _______ Direction__________

Vector # 2________ Direction__________

Vector Diagram:

115.8 m

227.2 m

South (-)

East (+)

Vr

V2

V1

Conventions:

+

+

-

-

1. Draw vector diagram. (Draw axis)

2. Resolve vectors into components using trig:

Vadj = V cos θ Vopp = V sin θ

3. Sum x and y components:

ΣVxi ΣVyi

4. Redraw!! Determine resultant vector using Pythagorean’s Theorem and trig:

Magnitude= √(Σ Vxi)² + (Σ Vyi)²

Direction Action: θ = tan-1(opp/adj)

Practice: Find FR = Fnet =?

200 N due South, 100 N at 40º N of W

Answer: Fnet = N @ ˚ W of S

Skip

An airplane flies at an engine speed of 100 m/s at 50º W of S into a wind of 30 m/s at 200 E of N. What is the airplane’s resultant velocity?

Solve using the components method!!

How far has the plane traveled after 1 hr?

a) Km b) Mph

Answer: 75.52 m/s @ 28.54˚ S of W

271 km or 168.89 miles per 1 hour

You Try!!!You Try!!!

A motor boat traveling 4.0 m/s, East encounters a current traveling 3.0 m/s, North.

a. What is the resultant velocity of the motor boat?

b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?

c. What distance downstream does the boat reach the opposite shore?

Another look, from a different perspective

Non-Collinear VectorsWhen 2 vectors are perpendicular, you must use the

Pythagorean theorem.

95 km,E

55 km, N

Start

Finish

A man walks 95 km, East then 55 km, north. Calculate his RESULTANTDISPLACEMENT

.The hypotenuse in Physics is

called the RESULTANT.

The LEGS of the triangle are called the COMPONENTS

Horizontal Component

Vertical Component kmc

c

bacbac

8.10912050

5595Resultant 22

22222

BUT……what about the direction?

In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer.

NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE.

N

S

EW

N of E

E of N

S of W

W of S

N of W

W of N

S of E

E of S

N of E

BUT…..what about the VALUE of the angle???

Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle.

N of E

55 km, N

95 km,E

To find the value of the angle we use a Trig function called TANGENT.

30)5789.0(

5789.095

55

1

Tan

sideadjacent

sideoppositeTan

109.8 km

So the COMPLETE final answer is : 109.8 km, 30 degrees North of East

What if you are missing a component?Suppose a person walked 65 m, 25 degrees East of North. What were

his horizontal and vertical components?

65 m25

H.C. = ?

V.C = ?

The goal: ALWAYS MAKE A RIGHT TRIANGLE!

To solve for components, we often use the trig functions since and cosine.

EmCHopp

NmCVadj

hypopphypadj

hypotenuse

sideopposite

hypotenuse

sideadjacent

,47.2725sin65..

,91.5825cos65..

sincos

sinecosine

ExampleA bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he

wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.

35 m, E

20 m, N

12 m, W

6 m, S

- =23 m, E

- = 14 m, N

23 m, E

14 m, N

3.31)6087.0(

6087.23

14

93.262314

1

22

Tan

Tan

mR

The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST

R

ExampleA boat moves with a velocity of 15 m/s, N in a river which flows

with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.

15 m/s, N

8.0 m/s, W

Rv

1.28)5333.0(

5333.015

8

/17158

1

22

Tan

Tan

smRv

The Final Answer : 17 m/s, @ 28.1 degrees West of North

ExampleA plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the

plane's horizontal and vertical velocity components.

63.5 m/s

32

H.C. =?

V.C. = ?

SsmCVopp

EsmCHadj

hypopphypadj

hypotenuse

sideopposite

hypotenuse

sideadjacent

,/64.3332sin5.63..

,/85.5332cos5.63..

sincos

sinecosine

ExampleA storm system moves 5000 km due east, then shifts course at 40 degrees

North of East for 1500 km. Calculate the storm's resultant displacement.

NkmCVopp

EkmCHadj

hypopphypadj

hypotenuse

sideopposite

hypotenuse

sideadjacent

,2.96440sin1500..

,1.114940cos1500..

sincos

sinecosine

5000 km, E

40

1500 km

H.C.

V.C.

5000 km + 1149.1 km = 6149.1 km

6149.1 km

964.2 kmR

91.8)364.0(

157.01.6149

2.964

14.62242.9641.6149

1

22

Tan

Tan

kmR

The Final Answer: 6224.14 km @ 8.91 degrees, North of East

Add vectors that are NOT perpendicular

• If the original displacement vectors do not form a right triangle– 1. Resolve each vector into its x- and y-

components

– 2. Find the sum of the x- and y-components

– 3. Use the Pythagorean Theorem to find the magnitude of the resultant

– 4. Use the tangent function to find the direction of the resultant

Adding non-perpendicular vectors

Adding non-perpendicular vectors

Adding non-perpendicular vectors

Practice #1

• A hiker walks 27.0 km from her base camp at 35 south of east. The next day, she walks 41.0 km in a direction 65 north of east and discovers a forest ranger’s tower. Find the magnitude and direction of her resultant displacement between the base camp and the tower.

Answer ~ 45 km at 29 degrees N of E