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Page 1: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

2-D motion

Page 2: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

2

Scalars and Vectors

• A scalar is a single number that represents a magnitude– Ex. distance, mass, speed,

temperature, etc.

• A vector is a set of numbers that describe both a magnitude and direction– Ex. velocity (the magnitude of velocity

is speed), force, momentum, etc.

Page 3: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

3

Scalars and Vectors

•Notation: – a vector-valued variable will be

Bold, – a scalar-valued variable will be in

italics. – if hand written vectors can be

denoted by an arrow over the value.

a

Page 4: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

4

Characteristics of Vectors

A Vector is something that has two and only two defining characteristics:

1. Magnitude: the 'size' or 'quantity'

2. Direction: the vector is directed from one place to another.

Page 5: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

A point at the beginning and an arrow at the end. The length of the arrow corresponds to the magnitude of the vector.•The direction the arrow points is the vector direction.Vectors are drawn to scale!!

Vectors can be drawn

Page 6: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

6

Example

•The direction of the vector is 55° North of East

•The magnitude of the vector is 2.3.

Page 7: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

7

Now You Try

Direction:

Magnitude:

47° North of West

2

Page 8: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

8

Try Again

Direction:

Magnitude:

43° East of South

3

Page 9: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

9

Try Again

It is also possible to describe this vector's direction as 47 South of East.

Why?

Page 10: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

There are two ways of reading a vector’s direction.

By comparing to a cardinal directionBy easting convention

Reading directions

Page 11: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

expressed as an angle of rotation of the vector about its tailEx. 40 degrees North of West

(a vector pointing West has been rotated 40 degrees towards the northerly direction)

Ex. 65 degrees East of South(a vector pointing South has been rotated 65 degrees towards the easterly direction)

By comparison

Page 12: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

a counterclockwise angle of rotation of the vector about its tail from due East.Ex. 30 degrees, 240 degrees

Easting Convention

Page 13: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

There are two ways of adding vectorsgraphicallyAnalytically

Resultant - the vector sum of two or more vectors. It is the result of adding two or more vectors together.

Vector addition

Page 14: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Graphic Addition Graphic Addition

Head-to-Tail MethodHead-to-Tail Method1. Draw the first vector with the proper length and orientation.

2. Draw the second vector with the proper length and orientation originating from the head of the first vector.

3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector.

4. Measure the length and orientation angle of the resultant.

Page 15: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m

Page 16: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Graphic Addition Graphic Addition •Ex. 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m

Page 17: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Graphic Addition Graphic Addition •The order of addition doesn’t matter. The resultant will still have the same magnitude and direction.

Page 18: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

This works only if the two vectors are at a right angle.

Page 19: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

Page 20: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Pythagorean TheoremPythagorean Theorem

Practice A: 10km North plus 5 km West. What is the resultant vector?

Practice B: 30km West plus 40km South. What is the resultant vector?

Page 21: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Trigonometry:

Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA

Page 22: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Trigonometry

This works only if the two vectors are at a right angle.Remember: SOH CAH TOA

Page 23: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Trigonometry

Ex. Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.Remember: SOH CAH TOA

Page 24: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition Trigonometry:

Let’s try these togetherLet’s try these togetherBack to Practice A and Practice BPractice A: 10km North plus 5 km West. What is the resultant vector?Practice B: 30km West plus 40km South. What is the resultant vector? Remember: SOH CAH TOA

Page 25: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Try this…Try this…Remember: SOH CAH TOAA plane travels from Houston, Texas to Washington D.C., which is 1540km east and 1160Km north of Houston. What is the total displacement of the plane?

Answer:1930 km at 37° north of east

Page 26: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Try this…Try this…Remember: SOH CAH TOAA camper travels 4.5km northeast and 4.5km northwest. What is the camper’s total displacement?

Answer:6.4km north

Page 27: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Try this…Try this…

Pg 89 Practice A #1-4

Page 28: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

28

Resolving Vectors/Expressing Vectors as Ordered Pairs

How can we express this vector as an ordered pair?

Use Trigonometry

These ordered pairs are called the components of the vector.

Page 29: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

29

Page 30: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

30

A good example:

Express this vector as an ordered pair.

Answer:(42.7, 34.6)

Page 31: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.

Answer:6.4km north

Page 32: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

Try this…Try this…Remember: SOH CAH TOAFind the components of the velocity of a helicopter traveling 95km/h at an angle of 35° to the ground.

Answer:y = 54km/hx = 78km/h

Page 33: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

One more…One more…Remember: SOH CAH TOAAn arrow is shot from a bow at an angle of 25° above the horizontal with an initial speed of 45m/s. Find the horizontal and vertical components of the arrow’s initial velocity.

Answer:41m/s, 19m/s

Page 34: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

Try this…Try this…

Pg 92 Practice B #1-4

Page 35: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving VectorsResolving Vectors

What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.

North39East

s

m47 o

Page 36: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving VectorsResolving Vectors

What if the vectors aren’t at right angles?Resolving a vector is breaking it down into its x and y components.First, we need a vector.

North39East

s

m47 o

Page 37: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

39o

E

Let’s draw the vector

s

m47

Page 38: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Continuing

Next we will draw in the component vectors which we are looking for.

Page 39: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Horizontal

Drawing the Components

Vertical

39o

s

m47

Page 40: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Identifying the Sides

Vertical

opphyp

adj

39o

s

m47

Horizontal

Page 41: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

What Trig Function will give the Horizontal Component?

opphyp

adj

39o

s

m47

Vertical

Horizontal

coscos adj

hyp

Page 42: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Finding The Horizontal Component

adjm

so 47 39cos

adjm

s36 5.

hyp

adjcos adj

hyp

adjcos coshypdja

hyp

adjcos

Page 43: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

opphyp

adj

Finding The Vertical Component

39o

s

m47

Vertical

Horizontal

sinsin opp

hyp

Page 44: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

oppm

so 47 39sin

Finding The Vertical Component

oppm

s 29 65.

hyp

oppsin opp

hyp

oppsin sinhypopp

hyp

oppsin

Page 45: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Continuing

The two components are:x:36.6m/sy: 29.65m/s

Page 46: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors PracticeResolving Vectors PracticeA plane takes off at a 35° ascent with a velocity of 195 km/h. What are the horizontal and vertical components of the velocity?

A child slides down a hill that forms an angle of 37° with the horizontal for a distance of 24.0 m. What are the horizontal and vertical components?

How fast must a car travel to stay beneath an airplane that is moving at 105 km/h at an angle of 33° to the ground (What is the horizontal component?) What is the vertical component of the plane’s velocity?

Page 47: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

More practice… More practice…

Pg 92 Practice B #1-4

Page 48: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Page 49: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Page 50: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

Page 51: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

What if the vectors aren’t at right angles?There are four steps.1.We have to resolve the vectors into their components.2.Add all the x components.3.Add all the y components. 4.Find the magnitude and direction of the resultant.

Page 52: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

A hiker walks 27km from her camp at 35° south of east. The next day, she walks 41km at 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:45km at 29° north of east

Page 53: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Analytically Addition Addition

A camper walks 4.5km at 45° north of east then 4.5 km due south. Find the camper’s total displacement, including direction.

Answer:3.4km at 22° south of east

You try…You try…

Page 54: 2-D motion. 2 Scalars and Vectors A scalar is a single number that represents a magnitude –Ex. distance, mass, speed, temperature, etc. A vector is a

Resolving Vectors

More practice… More practice…

Pg 94 Practice C #1-4


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