• Vectors have magnitude AND direction. – (14m/s west, 32° and falling [brrr!])

• Scalars do not have direction, only magnitude. – ( 14m/s, 32° )

Vectors

tip (head)tail

Vectors are written as: a , b , c, etc.. Or a , b , c, etc..

Vectors can be used to graphically describe an object’s motion.

We use arrows to represent vectors.

Directions:

Go 4 blocks SouthTurn and go 4 blocks West.

N

S

EW

Vectors can help us find our way – like directions.

Image you go to a party. You get directions from your house to the party.

N

S

EW

Directions:

Go 3 blocks SouthTurn and go 4 blocks West.

GOOD

N

S

EW

BETTER BEST

Directions:

Go 5 blocks South West.

Distance = _________

Distance = _________Displacement = _______

Directions:

Go 3 blocks EastTurn and go 3 blocks SouthTurn and go 7 blocks West.

Vectors

The longer the arrow, the larger the magnitude. The head of the arrow indicates the direction of travel.

Here are examples:

N

S

EW

58

1

On map coordinates: 1 unit of motion South 5 units of motion East 8 units of motion West

+5-8

0

On x-axis coordinates: 5 units of motion right 8 units of motion left

For example, which makes more sense?

A:

B:

The length of the arrow must be an indication of the magnitude of the vector.

Vector Magnitude

11m 16m 3m 9m

9m 3m 16m 11m

Vector Magnitude & Scale

By using a scale, we can assign values to the length of a drawn vector.

For example: if the scale is 1cm = 5m/s

then a vector that is drawn 3cm long would represent 15m/s.

Knowing this, can you calculate the magnitude of the following vectors?

ABC

Scale: 1cm = 5 m/s

The direction of the arrow head indicates the direction of travel. We will use many coordinate systems, but for now, let’s stick to a map

system (N,S,E,W)

Vector Direction

N

S

EW

This red arrows represents a vector.

It represents motion in the East direction.

N

S

EW

This green arrows represents a vector.

It represents motion in the South direction.

Which vector represents faster motion?

N

S

EW

Vector Direction

Not all vectors will lie along an axis. We must then specify the angle and direction of the vector.

35°

This vector is 35° North of East

This vector is 27° East of South

27°

We can graphically find the angle of the vector by using a protractor.

N

S

EW35°

This vector is 35° North of East

This vector is 27° East of South27°

Naming a vector direction can be very tricky . . . Be very careful!

This vector is 35° East of North

35°

27°This vector is 27° South of East

Graphically Analyzing Vectors

N

S

EW27°

We can analyze vectors by looking at their Magnitude and Direction.

Magnitude: use a ruler and measure the length of the arrow. Use the scale to calculate the magnitude of the vector.

Direction: use a protractor to measure the angle of the arrow relative to a coordinate axis. Give the full direction with respect to that axis.

N

S

EW27°

Scale: 1cm = 10m/s

Write out the full vector: 70m/s @ 27° North of East70m/s @ 63° East of North

OR

N

S

EW

Vector Resolution

Vectors are not always on the coordinate axis. When this occurs, we need to break down the vector into its x component and its y component. This is called vector resolution.

X component

Y component

To do this:1. draw a vertical line at the head of the vector parallel to the vertical axis.2. draw a horizontal line at the head of the vector parallel to the horizontal axis.

Measure to find the magnitude !

Finding the Resultant Vector

N

S

EWX component

Y componentMeasure to find the magnitude !

Sometimes you may know the X component and Y component of an object’s motion.In this case, you must take those two vectors and substitute them with one vector.This vector is called the Resultant Vector. A Resultant vector is the simplest vector that allows you to represent the same motion.

To do this: 1. draw a vertical line at the head of the X component, parallel to the vertical axis. 2. draw a horizontal line at the head of the Y component, parallel to the horizontal axis. 3. draw the Resultant Vector from the tails of the components to the intersection of the dotted lines.

Measure to find the direction !

Vectors can be moved anywhere on your page.

All of these vectors have the same magnitude and direction.

If you move a vector on your page, it does not change as long as The length and direction of the arrow remains the same.

Not the same vectorWHY ??

Vector Addition

When we add vectors, we just put the tail of one vector onto the head of the other vector.

Here’s an example:trip #1: a car travels 30m left.trip #2: a car travels 20m up.trip #3: a car travels 40m right.

how can we add the car’s vectors?

This is the:

Tail-to-Tip Method

Vector Addition

Let’s look at some linear vector addition: (in a straight line)

If a plane is flying at 100m/s East and encounters a tail wind of 20m/s, how fast is the airplane going?

100 m/s20 m/s

By vector addition:

120 m/s(Tail-to-Tip Method)

This is called the Resultant Vector- It goes from the tail of the first vector to the tip of the last vector

Vector Addition

Let’s look at some linear vector addition: (in a straight line)

If a plane is flying at 100m/s East and encounters a head wind of 20m/s, how fast is the airplane going?

100 m/s 20 m/s

By vector addition:

80 m/s

This is called the Resultant Vector- It goes from the tail of the first vector to the tip of the last vector

(Tail-to-Tip Method)

Vector Addition

Now let’s look at some vector addition NOT in a straight line.

If a plane is flying at 100m/s East and encounters a crosswind of 20m/s, how fast is the airplane going?

100 m/s

20 m/s

By vector addition:

100 m/s

20 m/s

You need Pythagorean TheoremTo solve this problem !

Drawn from tail of first vector

to head of second vector

a2 + b2 = c2

1002 + 202 = c2

10400 = c2

102m/s = c

c

(Tail-to-Tip Method)

tail wind head wind cross wind

This is the “real life” effect of wind vectors on an aircraft:

Called “wind shear” - this is very dangerous to pilots.

This is why it takes about8 hours to fly from PHL to LAX

This is why it takes about 6 hours to fly from LAX to PHL

Vector Addition

This is vector addition

It is written as: a + b or as a + b

(Tail to Tip Method) Resultant vector

a

b

???? Is a + b = b + a ????Riddle me this Bartman . . .

a

b

a

b

YES, it has the same Resultant vector !

a + b

b + a

Opposite of a Vector

The opposite of a vector has the same magnitude as a given vector, but opposite direction.

Definition:

N

S

EW

5 m/s

Original vector

Opposite vector5 m/s

The opposite of vector A is vector -A

For example: A is 12m East, -A is 12m West

12mEW

-A

A

-A

A12m

Subtracting Vectors

Subtracting vectors is the same as adding an opposite of a vector.

To calculate: A – B

Turn it into addition of an opposite of a vector: A + (-B)

Add as usual.

Example: Vector A: 10m/s @ EVector B: 7m/s @ N

A

BA+B

A

-BA-B

Multiplying Vectors

Vectors can be multiplied by a scalar value. - The direction will NOT change but the magnitude will.

You drive at a constant velocity Eastward of 30m/s for 4s. What is your displacement?Example:

vector scalar

Mathematically: displacement = ∆d = v * ∆t∆d = 30 * 4∆d = 120m

vector

Graphically:

E

30m/s 60m/s 90m/s 120m/s

Working with more than 2 vectors:

By using the Tail-to-tip Method, you could graphically add two or more vectors.

AB

C+ + =

A

B C

We can add as many vectors together as we wish. But, when we want to calculate the Resultant Vector, we must remember that it is the vector going FROM the Tail of the FIRST vector TO the Head of the LAST vector.

A

BC

Resultant Vector

Tail of FIRST vector

Head of LAST vector

SO:A

BC

+ + = R

Examples of Graphical Vector Solutions using Tail-to-Tip Method and Resultant Vectors:

N

S

EW

A commercial airliner flies 100m East from PHL to Atlantic City Airport. It then flies 1000m South to Charlotte SC. It then flies 2000m West to Las Vegas. It makes its last leg of the trip by flying 2800m North to Green Bay WI. What is the airliner’s displacement?

Graphically: The Resultant vector is the displacement.

Tip:

Use a scale to measure the length of the Resultant Vector to obtain the magnitude.Use a protractor to measure the direction of the Resultant Vector.

Working with X and Y Components

When adding two or more vectors, break each down into its X Component and its Y Component. Then add your X’s and your Y’s. Use these two values to compute the Resultant Vector:

A

6

4

B

1

7

A+ B: 6 4

-1 -7

5 -3

X

X

YY

A

B

R

5

-3X

YX comp Y comp

Using Trigonometry to Analyze Vectors

X

YEven though you can solve for magnitude and direction by using a ruler and protractor, it is nice to be able to use equations also.

Trig Equations:

a2 + b2 = c2 Pythagorean’s Theorem

sin (Θ) = opposite / hypotenusecos (Θ) = adjacent / hypotenusetan(Θ) = opposite / hypotenuse

Note:If you are using the angle to find a side of the triangle, you will use the trig functions sine, cosine, and tangent.

If you are using the sides of the triangle to find the angle, you will use the trig functions, sin-1, cos-1, and tan-1.

opp

adj

hyp

Θ

SohCahToa