Scalars & Vectors

Vectors are quantities that have both a direction

and a magnitude (size).

Ex. 2 km, 30ο north of east

Examples of Vectors used in Physics

Displacement

Velocity

Acceleration

Force

Scalars are quantities that have only a

magnitude(size) are called.

Scalar Example Magnitude

Speed 20 m/s

Distance 10 m

Age 15 years

Heat 1000 calories

Vectors can be represented by words “Take your team 2 ‘clicks’ (km) north”

“US Air 45, new course 30o at 500 mph.”

Vectors can be represented by symbols In the text, boldface indicates vectors.

Examples:

Vectors can be represented graphically using arrows The direction of the arrow is the direction of the

vector. The length of the arrow tells the magnitude

Vectors can be moved parallel to themselves and still be the same vector

Vectors only tell amount and direction, so a vector doesn’t care where it starts.

tΔx

VaF av

The sum of two vectors is called the

resultant.

To add vectors graphically, draw each vector

to scale.

Place the tail of the second vector at the tip

of the first vector.

Vectors can be added in any order.

To subtract a vector, add its opposite.

VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them.

Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started?

54.5 m, E 30 m, E +

84.5 m, E

Notice that the SIZE

of the arrow conveys

MAGNITUDE and the

way it was drawn

conveys DIRECTION.

VECTOR SUBTRACTION - If 2 vectors are going in

opposite directions, you SUBTRACT.

Example: A man walks 54.5 meters east, then 30

meters west. Calculate his displacement relative to

where he started?

54.5 m, E

30 m, W

24.5 m, E

When 2 vectors are perpendicular, you must use the Pythagorean Theorem.

kmc

c

bacbac

8.10912050

255295Resultant

22222

95 km,E

55 km, N

Start

Finish The hypotenuse in Physics is

called the RESULTANT.

The LEGS of the triangle are called the COMPONENTS

Horizontal Component

Vertical

Component

Example: A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT

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

E W

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

Just putting North of East on the answer is NOT specific enough for

the direction. We MUST find the VALUE of the angle.

o30)5789.0(1

5789.09555

Tan

sideadjacentsideoppositeTan

95 km, E

To find the value of the

angle we use a Trig

function called TANGENT.

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

N of E

55 km, N

109.8 km

Resolve each vector into x and y

components, using sin and cos.

Add the x components together to get the

total x component. Add the y component

together to get the total y component.

Find the magnitude of the resultant using

Pythagorean theorem.

Find the direction of the resultant using the

inverse tan function.

Any vector can be resolved, that is, broken

up, into two vectors, one that lies on the x-

axis and one on the y-axis.

An arrow is shot from a bow at an angle of 25ο

above the horizontal, with an initial speed of 45

m/s. Find the horizontal and vertical components

of the arrow’s initial velocity.

25o

vx

vy

?

?

25

m/s 45

y

x

v

v

v

o

m/s 4178.40

)cos(25m/s) 45(

cos

cos

x

x

x

x

v

v

vv

v

v

m/s 1901.19

)(25sinm/s) 45(

sin

sin

y

y

y

y

v

v

vv

v

v

Suppose a person walked 65 m, 25 degrees East of North.

What were his horizontal and vertical components?

EmCHopp

NmCVadj

hypopphypadj

hypotenuse

sideopposite

hypotenuse

sideadjacent

,47.2725sin65..

,91.5825cos65..

sincos

sincos

65 m 25

H.C. = ?

V.C = ?

The goal: ALWAYS MAKE A RIGHT

TRIANGLE!

To solve for components, we often use

the trig functions since and cosine.

A 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.

3.31)6087.0(

6087.23

14

93.262314

1

22

Tan

Tan

mR

35 m, E

20 m, N

12 m, W

6 m, S

- = 23 m, E

- = 14 m, N

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

23 m, E

14 m, N R

A 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.

1.28)5333.0(

5333.015

8

/17158

1

22

Tan

Tan

smRv

15 m/s, N

8.0 m/s, W

Rv

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

A 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.

SsmCVopp

EsmCHadj

hypopphypadj

hypotenuse

opposite

hypotenuse

adjacent

,/64.3332sin5.63..

,/85.5332cos5.63..

sincos

sinecos

63.5 m/s

32

H.C. =?

V.C. = ?

A 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

opp

hypotenuse

adj

,2.96440sin1500..

,1.114940cos1500..

sincos

sinecosine

0.20)364.0(

364.01.2649

2.964

1.28192.9641.2649

1

22

Tan

Tan

kmR

5000 km, E

40

1500 km

H.C.

V.C.

1500 km + 1149.1 km = 2649.1 km

2649.1 km

964.2 km R

The Final Answer: 2819.1 km @ 20

degrees, East of North

We use the term VECTOR RESOLUTION to suggest

that any vector which IS NOT on an axis MUST be

broken down into horizontal and vertical

components.

BUT --- the ultimate and

recurring themes in

physics is take any and all

vectors and turn

them all into ONE BIG

RIGHT TRIANGLE.

1. Make a drawing showing all the vectors, angles, and given directions.

2. Make a chart with all the horizontal components in one column and all the vertical components on the other.

3. Make sure you assign a negative sign to any vector which is moving WEST or SOUTH.

4. Add all the horizontal components to get ONE value for the horizontal. Do the same for the vertical.

5. Use the Pythagorean Theorem to find the resultant and Tangent to find the direction.

A search and rescue operation produced the

following search patterns in order:

1: 30 meters, west

2: 65 meters, 32 degrees East of South

3: 130 meters, east

4: 42 meters, 22 degrees West of North

1: 30 meters, west

2: 65 meters, 32

degrees East of

South

3: 130 meters, east

4: 42 meters, 22

degrees West of

North

130 m, E

30 m, W

65 m

32ο 42 m

22ο

1: 30 meters, west

2: 65 meters, 32

degrees East of South

3: 130 meters, east

4: 42 meters, 22

degrees West of

North

Leg Horizontal Vertical

1 30 m 0 m

2

3 130 m 0 m

4

65 meters, 32 degrees

East of South

Leg Horizontal Vertical

1 30 m 0 m

2

3 130 m 0 m

4

65 m 32

h.c.

v.c

65cos32 =

55.12 m

65sin32 =34.44 m

55.12 m 34.44 m

42 meters, 22 degrees

West of North

Leg Horizontal Vertical

1 30 m 0 m

2 34.44 m 55.12 m

3 130 m 0 m

4

42 m 22 v.c

h.c. 42sin22 =15.73 m

15.73 m

42cos22=38.94 m

38.94 m

1: 30 meters, west

2: 65 meters, 32

degrees East of

South

3: 130 meters, east

4: 42 meters, 22

degrees West of

North

Leg Horizontal Vertical

1 30 m 0 m

2 34.44 m 55.12 m

3 130 m 0 m

4 15.73 m 38.94 m

-

-

-

1: 30 meters, west

2: 65 meters, 32

degrees East of South

3: 130 meters, east

4: 42 meters, 22

degrees West of

North

Leg Horizontal Vertical

1 -30 m 0 m

2 34.44 m -55.12 m

3 130 m 0 m

4 -15.73 m 38.94 m

Total 118.71 m -16.18 m

What does this mean??? 118.71 m

-16.18 m

2 2118.71 ( 16.18) 119.81R m

1

16.180.136

118.71

(0.136) 7.76

Tan

Tan

65 m

32

130 m, E

42 m 22

30 m, W 118.71 m

-16.18 m

Final Answer: 119.81 m, 7.76 degrees, South of East