3.1 & 3.2 Vectors & Scalars

Biblical ReferenceThe Lord will grant that the enemies who rise up against you will be defeated before you. They will come at you from one direction but flee from you in seven.

Deuteronomy 28:7

Scalar

Scalar Example Magnitude

Speed 20 m/s

Distance 10 m

Age 15 years

Heat 1000 calories

Any quantity in physics that has Magnitude, but No Direction

Magnitude: A numerical value with units.

Vector

Vector Magnitude & Direction

Velocity 20 m/s, N

Acceleration 10 m/s/s, E

Force 5 N, West

Faxv

,,, Click Here for a better explanation of vectors.

Any quantity in physics that has Both Magnitude and Direction

• Typically illustrated with an arrow above the symbol to convey magnitude & direction

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?

Applications of Vectors

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.

Resultant – A vector that represents the sum or 2 or more vectors

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?

Applications of Vectors

54.5 m E

30 m W-

24.5 m E

The addition of another axis helps to describe motion.

Two Dimensional Coordinate Systems

Non-Collinear Vectors

kmc

c

bacbac

8.10912050

5595Resultant 22

22222

When 2 vectors are perpendicular, use the Pythagorean Theorem.

95 km E

55 km N

Start

FinishExample: A man walks 95 km E then 55 km N. Calculate his Resultant Displacement.

The hypotenuse is called the Resultant.

The Legs of the triangle are called the Components.

Horizontal Component

Vertical Component

You should also include a Direction for all vectors.

BUT……what about the direction?

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?

30)5789.0(

5789.095

55

1

Tan

sideadjacent

sideoppositeTan

Just putting North of East on the answer is Not specific enough for the direction. You Must find the Value of the angle.

N of E

55 km N

95 km E

To find the value of the angle, use a Trig function called Tangent.

q

109.8 km

The Complete final answer is : 109.8 km, 30 North of East

Example – Find the ResultantWhile following the directions on a treasure map, a pirate walks 45.0 m north and then turns and walks 7.50 m east. What is the single straight line displacement?

45 m N

7.5 m E

mc

c

bacbac

6.4525.2081

5.745Resultant 22

22222

46.9)1667.0(

1667.045

5.7

1

Tan

sideadjacent

sideoppositeTan

East of North

or 80.54 North of East

What if you are missing a component?

mECHopp

mNCVadj

hypopphypadj

hypotenuse

sideopposite

hypotenuse

sideadjacent

47.2725sin65..

91.5825cos65..

sincos

sinecosine

Suppose a person walked 65 m, 25 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, use the trig functions sine and cosine.

Example – Find components

uphrkmCVopp

awayhrkmCHadj

hypopphypadj

hypotenuse

opposite

hypotenuse

adjacent

,/49.5435sin95..

,/82.7735cos95..

sincos

sinecosine

A helicopter flies at 95 km/hr at an angle of 35 to the ground. Find the components of its motion.

95 m

35

H.C. = ?

V.C = ?

ground

Example

1.28)5333.0(

5333.015

8

/17158

1

22

Tan

Tan

smRv

A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s W. Calculate the boat's resultant velocity with respect to due north.

15 m/s, N

8.0 m/s, W

Rv

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

Example

SsmCVopp

EsmCHadj

hypopphypadj

hypotenuse

opposite

hypotenuse

adjacent

,/64.3332sin5.63..

,/85.5332cos5.63..

sincos

sinecosine

A plane moves with a velocity of 63.5 m/s at 32 South of East. Calculate the plane's horizontal and vertical velocity components.

63.5 m/s

32

H.C. =?

V.C. = ?

Example

NkmCVopp

EkmCHadj

hypopphypadj

hyp

opp

hyp

adj

,2.96440sin1500..

,1.114940cos1500..

sincos

sinecosine

A storm system moves 5000 km due east, then shifts course at 40 N of E for 1500 km. Calculate the storm's resultant displacement.

9.8)157(.

157.01.6149

2.964

2.62242.9641.6149

1

22

Tan

Tan

kmR

5000 km, E

40

1500 km

H.C.

V.C.

5000 km + 1149.1 km = 6149.1 km

6149.1 km

964.2 kmR

Final Answer: 6224.2 km @ 8.9 N of E

Adding Non-Perpendicular Vectors

• Many objects move in one direction and then turn at an angle before continuing their motion.

• Break the displacement vectors into x and y components.

Example – Non-Perpendicular VectorsA bus travels 301 m 23° above the x-axis. Then it turns and travels 235 m 12° above the x-axis. What is the displacement of the bus?

= +

A B

BATot XXX BATot YYY

Tot

Tot

X

YTan

Example – Non-Perpendicular Vectors

A23

301 m

NmY

EmX

hypYhypX

hyp

opp

hyp

adj

A

A

AA

,61.11723sin301

,07.27723cos301

sincos

sinecosine

Example – Non-Perpendicular Vectors

NmY

EmX

hypYhypX

hyp

opp

hyp

adj

B

B

BB

,86.4812sin235

,86.22912cos235

sincos

sinecosine

B

12235 m

Example – Non-Perpendicular Vectors

EmmmXXX BATot ,93.50686.22907.277

NmmmYYY BATot ,47.16685.4861.117

mYXR TotTot 53447.16693.506 2222

1893.506

47.16611 TanX

YTan

Tot

Tot

What if we subtract the second vector from the first in the previous problem?

A = 301 m 23° above the x-axis

B = 235 m 12° above the x-axis

A – B = A + (-B)

Example – Subtracting Two Vectors

A = 301 m 23° above the x-axis B = 235 m 12° above the x-axis

Last Example

NmCVopp

EmCHadj

hypopphypadj

hyp

opp

hyp

adj

,9.1527sin35..

,2.3127cos35..

sincos

sinecosine

A man walks 26 m East and then walks 35 m 27 North of East. What is his displacement?

5.15)364.0(

278.02.57

9.15

4.599.152.57

1

22

Tan

Tan

mR

26 m, E

27

35 m

H.C.

V.C.

26 m + 31.2 m = 57.2 m

57.2 m

15.9 mR

q

Final Answer: 59.4 m @ 15.5 N of E