Vectors and ScalarsVectors and Scalars

A SCALAR is ANY quantity in physics that has MAGNITUDE, but NO direction.

Scalar Example

Magnitude

Speed 20 m/s

Distance 10 m

Age 15 years

Heat

Number of horses

behind the school

1000 calories

I guess: 12

V

A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION.

Vector Magnitude & Direction

Displacement 5 m, NWVelocity 20 m/s, NAcceleration 10 m/s/s, EForce 5 N, West

Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

How are velocity and speed related?

Speed is a scalar – it just has magnitude. Velocity has both magnitude and direction.

Example: 20 m/s = speed 20 m/s NE = velocity

A VECTOR is ANY quantity in physics that has BOTH MAGNITUDE and DIRECTION.

Vector Magnitude & Direction

Displacement 5 m, NWVelocity 20 m/s, NAcceleration 10 m/s/s, EForce 5 N, West

Vectors are typically illustrated by drawing an ARROW above the symbol. The arrow is used to convey direction and magnitude.

v

Any guesses as to what displacement is? (Hint, look at the units!)

Displacement is the vector quantity of distance … that is, it tells how far and in what direction two things are located.

The length of the vector, drawn to scale, indicates the magnitude of the vector quantity.

the direction of a vector is the counterclockwise angle of rotation which that vector makes with due East or x-axis.

How to draw vectors

Tail

HeadL

H

250

x

length = magnitude 6 cm

250 above x-axis = direction

displacement x = 6 cm, 250

How to draw vectors – lady bug displacement

NOTE 2:

When drawing a vector you MUST MUST MUST put a ‘head’ or arrow on the vector to demonstrate which direction it is pointing.

NOTE 1:

Displacement shows how far apart something is now compared to from where it started. It does not show show the path or the total distance travelled.

QUICK REVIEWWhat is the difference between a scalar and a vector?

What are the parts of a vector?

QUICK REVIEWWhat is the difference between a scalar and a vector?

Scalars have magnitude only, vectors have magnitude and direction.

What are the parts of a vector?

tail head

A resultant (the real one) velocity is sometimes the result of combining two or more velocities.

A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it)

1. The plane encounters a tailwind of 80 km/h.

The resulting velocity relative to the ground is 280 km/h S

e

80200 kmh

kmh

280 kmh

Adding Vectors – Plane example 1 (tailwind)

To understand how far the plane is traveling relative to the ground we need to add the two vectors – the plane’s heading and the tailwind.

We add vectors by moving them head to tail and finding the resultant (sum).

200kmh

80 km/ h

A small plane is heading south at speed of 200 km/h (This is what the plane is doing relative to the air around it)

2. It’s Texas: the wind changes direction suddenly 1800. Now the plane encounters a 80 km/h headwind

The resultant velocity is only 120 km/h south..

e 200 km

h

80 kmh

120 kmh

Adding Vectors – Plane example 2 (headwind)

How do we figure out the plane’s velocity relative to the ground?

Move the vectors head-to-tail and find the resultant vector.

The resultant vector always goes from the tail of the first vector to the head of the second vector

Adding or subtracting vectors in a straight line is easy, but what if the wind is coming from the side?

We need to use trigonometry.

3. The plane encounters a crosswind of 80 km/h.Will the crosswind speed up the plane, slow it down, or have no effect? How can we find out?

As in the other two examples, we have to add two velocity vectors head-to-tail in order to find the

resultant vector.

200 kmh

80 kmh

RESULTANTRESULTANT VECTOR(RESULTANT VELOCITY)

How can we calculate the magnitude of the resultant vector?

Use trigonometry! Specifically, Pythagorean theorem.

v = 215 km/h SEHow can I find the exact angle?

Use trig! Specifically, tan-1.

2 2 2 2 2 21 2v= v +v = (200km/h) + (80km/h) = 46400km /h

80kmh

200 kmh

F = tan-1 (80 / 200) = 22o

So the plane is traveling 215 km/h at 22o E of S.

Because direction matters!

200 kmh 280 km

h 120 kmh 215 km

h 180 kmh

So why do we use vectors in physics???

vectors can be moved around as long as their length (magnitude) and direction are not changed.

Vectors that have the same magnitude and the same direction are the same.

The order in which two or more vectors are added does not effect result.

Adding A + B + C + D + E yields the same result as adding C + B + A + D + E or D + E + A + B + C.

Quick check:

Are all these vectors equal or not?

How do you know?

Yes! They are all equal. A vector quantity is determined by its length and direction. Its position doesn’t matter. That’s why we can move vectors around to add them.

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, W24.5 m, E

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

54.5 m, E 30 m, E

84.5 m, E

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

54.5 m, E

30 m, N62.2 m, NE

2 254.5 + 30.0 = 62.2

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.

q

So the COMPLETE final answer is :

54.5 m, E

30 m, N

62.2 m, NE30

θ = arc tan 54.5

= 290

62.2 m, 290 or 62.2 m @ 290

Try the following on your own A person walks 5m N then walks 8m S.

Calculate their displacement.

A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity.

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

Try the following on your own. A person walks 5m N then walks 8m S.

Calculate their displacement.

A ball is thrown 25 m/s E. A tailwind of 5 m/s E is blowing. Calculate the resulting velocity.

5 m 8 m

3 m 3m South

5 m 25 m

30 m 30m East

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.

28.1(0.5333)][tan(0.5333)arctanθ

0.533315

8θtan

m/s17158v

1

22

15 m/s, N

8.0 m/s, W

q

The Final Answer :

118.1 m/s,17

N ofW 28.1 @ m/s17

v

v

v

Example 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.

2 2R = 14 +23 = 26.93m

35 m, E

20 m, N

12 m, W

6 m, S

23 m

14 m

The Final Answer:

Rq

14tanθ = = 0.6087

23

-1 oθ = tan (0.6087) = 31.3

IMPORTANT NOTE:

The Pythagorean theorem and trig functions can only be used for right angle triangles!

Later, I will teach you how to handle vectors that meet at angles other than right angles.

1 2v + v = v

1v

2v

v

Multiplying vector by a scalar

Multiplying a vector by a scalar will ONLY CHANGE its magnitude – not direction.

Opposite vectors

One exception:Multiplying a vector by “-1” does not change the magnitude, but it does reverse it's direction

A 2A 3A ½ A

A

- A

– A

– 3A