What kinds of things can be represent by vector Displacement- Magnitude- how far you went Direction - which way Velocity -- Magnitude- speed Direction.

What kinds of things can be represent by vector

Displacement- Magnitude- how far you wentDirection - which way

Velocity -- Magnitude- speedDirection - which way

ForcesMagnitude- how hard you are pushing or pullingDirection - which way

Acceleration -- Magnitude- change in speedDirection - which way

Vectors can be represent several ways

2 m East 4 m West

x=+2 m x=-4 m

Words

Math

Pictures

The length of the arrow represents the magnitude(if a scale is given like 1 cm = 5 m/s)

The arrowhead points in the direction

GRAPHICALLY

Vectors contain 2 pieces of information How much & which way.

MOVING them does not change that information

2 m

This means that they can be moved and redrawn as long as that information does not change

Just as scalars can be added

3 apples + 5 apples = 8 apples

So can vectors ( but you need to consider )

3 m + 5 m doesn’t always = 8 m

WHY????

direction

To add 2 or more vectors (graphically) you put them head to tail

BAD

GOOD

The vector sum is called the

RESULTANT

resultant

It is an arrow drawn from the beginning of the first vector to the head of the final vector

Consider displacement

If you walk 2 m (right) then another 3 m (right)

The resulting displacement (called the resultant)is 5 m to the right

2 m 3 m

5 m (resultant)

2 m 3 m

5 m (resultant)

Using signs (+/-) to indicate direction

+2 m +3 m+ = +5 m

Graphically adding vectors

Mathematically adding vectors

(using + to mean right)

If you walk 2 m (right) then 3 m (left)

2 m

3 m

the resultant is 1 m (left)

1 m (resultant)

2 m

3 m

1 m (resultant)

2 m -3 m+ = -1 mnegative indicates left

Using signs (+/-) to indicate direction (using standard +x)

Graphically adding vectors

Mathematically adding vectors

These rules work with all vectors not just displacement.

Take velocity

A person walks 1 m/s N

A conveyor moves 2 m/s N

What is the resultant of adding the two vectors

3 m/s N

What does the resultant mean here?

Using signs

Take velocity

person’s velocity relative to walkway vy= +1 m/s

vy= +3 m/s

Walkway velocity relative to the ground vy= +2 m/s

Resultant

Person’s velocity relative to the ground vy= +2 m/s

Using signs

Take velocity

A person relative to walkway vy= +1 m/s

vy= -1 m/s

Walkway relative to groundvy= -2 m/s

Resultant

What is the resultant of adding the two vectors

Using velocity instead• When vectors point in the same direction

we add them just as we would add any two numbers.

When adding vectors (order does not matter)

+ =

start FinishA

B

A B

B A

Resultant

1 + 2 = 2 + 1

In other “words”

A + B = B + A

+

Start

Vectors can be added graphically just by putting themhead to tail (order does not matter)

A

B

(+2)

-1

Finish

Resultant

AB

A

B

2 + (-1) = (-1) + 2

In other “words”

A + B = B + A

Subtracting vectors just involves flipping the vector being subtracted and then adding like normal

2 m right 3 m right-means

2 m 3 m left+

minus

plus

Just like 2 + (-3) =

What will the answer be?

3 m 3 m left-

means

minus

+plus3 m 3 m right

6 m

Adding vectors by walking activity(Theoretically in the front of the school. Weather pending. All terms and conditions are subject to whatever the teacher feels like. Don’t make him crabby. Not valid in the state of California. Many will enter, few will win. Not valid with any other offer.)

In 2 dimensions, all the same rules apply

+

To add them just put them head to tail

2 m East

3 m North

Start with 1 vector

3 m North

In 2 dimensions, all the same rules apply

2 m East +

The RESULTANT (sum) is a vector drawn from start to finish

2 mEast

3 m North

3 m North

In 2 dimensions, all the same rules apply

2 m East +

We could have started with the green vector first

3 m

2 m

3 m North

In 2 dimensions, all the same rules apply

2 m East +

2 m

3 m

3 m North

The resultant is a vector drawn from start to finish

In 2 dimensions, all the same rules applyIn 2 dimensions, all the same rules apply

2 m East +

2 m

3 m

3 m North

2 m

3 m

The order doesn’t matter, because the resultant is the same!!!

2 m

3 m

The resultant is EQUIVALENT to the addition

resultant

If you had walked the path of the resultant you would be in the same place as walking the original vectors

Go back to the walkway idea… Adding velocity vectors??WHAT COULD THAT MEAN?????

What if you walked N on a walkway that was moving E. Which way would you go? And go compared to what?

Sorry, Office 2010 at home. ArrrgggHHH hard to use now. Not compatible with version here. So on this slide this is as good as I can do.

Its all relative(s) get it? (this is a random picture from the internet, none of these people are related to me)

Do the 3 examples below all show correct addition?

Vectors can only be added (head to tail)

A+ B C+

Which shows the vectors being added correctly?

1 2 3

4 5 6

1 2 5

NOTICE: the resultant is the same for all 3 done correctly

Add the two vectors

OR We could have started with the Green Vector First

Find the Resultant

Either way you do it the resultant is the same

Notice a parallelogram is formed

This is another way to add vector.Usually if the vectors are given tail to tail

Then complete the parallelogram

Then draw the resultant from start to finish

What is your resulting displacement vector after walking 15 m due south then 25 m due west

Pick a scale 1 dm = 5 m

Draw the vectors

Find the resultant of A + B

1 dm = 15 m/s

Draw the vectors

A

B

Find the resultant using the head to tail method. 6 miles W + 4 miles N – 4 miles W1 dm = 1 mile

Find the resultant using the head to tail method. 14 m 300 N of East + 24 m 300 W of North

1 dm = 4 m

Read section 3-4

• How big is the resultant?

We could get out a ruler, but there is a better way!!

Remember This?

• How big is the resultant?

Analytical Method of VectorAddition

• The sum of any two vectors can bedetermined using trigonometry.

What is the resultant velocity of a boat if it is the resultant of two vectors.

35 m/s due S and 12 m/s due E

You are initially driving 13 m/s due south. After turning for 12 seconds your velocity is 9 m/s due east. Find the average acceleration during this period. (hint: how would you find v?)

A car is driven 31 km East and 15 km due NE. What is the resultant displacement? (remember displacement is a vector so it should have magnitude and direction)

31 km

15 km

R

No problem use the Pythagorean Theorum, right??

31 km

15 kmR

NO, that only works for right triangles!!!!!

31 km

15 km

R

A2 + B2 = C2

A2 + B2 = C2

3 approaches for adding non-perpendicular vectors

Graphical Approach

Law of cosines

Addition by vector resolution

Works with any number of vectors, but highest error

Accurate, but only works with two vectors at a time. Can be the fastest but not always

Accurate, Most versatile. More work

A

B

R2 = A2 + B2 – 2ABCos()

31 km

15 km

R

R

But how to find the angle?

Graphing packet

We have been adding 2 vectors to form their resultant

X

YResultant =

A single vector can be thought of as the sum of 2 components in the x and y axes

AX

AY

A

A single vector can be thought of as the sum of its two component vectors.

AX

AY

A= +

A = AX + AY

Any Vector can be broken into X & Y Component VectorsSimply form a right triangle, putting proper directional arrow heads on the components

AAy

Ax

A = Ax Ay+

Component Vectors

Vector Components

• We can take two vectors and replace themwith a single vector that has the same effect.This is vector addition.

• We can start with a single vector and thinkof it as a resultant of two perpendicularvectors called components.

• This process is called vector resolution.

To get to your campsite you hiked, 23o N of east for 2.6 km.

Afterwards, what is your N/S displacement and your E/W displacement

Can we add these vectors

A B

Can we use Pythagorean’s Theorem?

Vectors that are not perpendicular (don’t form a right triangle)

can be added a different way.It involves breaking them up into their components 1st

A

B

Ax

Ay

Bx

By

Then do the same for the Y’s to find Ytot

Add up all the X’s to find Xtot

A

B

Ax

Ay

Bx

By

Ax

Bx

By

Ay

A

B

Ax

Ay

Bx

By

AxBx

By

Ay

Ax + Bx = Xtot

Ay + By = Ytot

The add Xtot & Ytot

To find the resultantWe will use Trigonometry to do this

A

B

Ax

Ay

Bx

By

Xtot

Ytot

1.) Label Vectors2.) Break apart each vector into its X & Y Components (watch signs before doing next step)3.) Add up all the X’s to find Xtot

4.) Then do the same for the Y’s to find Ytot

5.) Draw a triangle with Xtot & Ytot (also watching direction)6.) Find the magnitude using Pythagoreans theorem7.) Find the angle using trig (forget + / -’s here)

Add the following

A B

30o

110o

2.5 m 2.0 m+

Ax

Ay

Bx

BY

70o

Ax = 2.5 m * cos (30) = 2.2 m

Ay = 2.5 m * sin (30) = 1.3 m

Bx = 2.0 m * cos (70) = -.68 m

By = 2.0 m * sin (70) = 1.9 m

SKIP

A B

30o

110o

2.5 m 2.0 m+

Ax

Ay

Bx

BY

70o

2.2 m

1.3 m 1.9 m

-.68 m

NOW Find Xtot & Ytot

Xtot = 2.2 m + (-.68m) = + 1.5 m

Ytot = 1.3 m + 1.9 m = + 3.2 m

SKIP

Finally find the resultant of Xtot & Ytot

Xtot = + 1.5 m

Ytot = + 3.2 m

SKIP

Find A + B + C & A + B - C

75o

66o 15 m

12 m

23 m

A

B

C

A plane’s engines propel it north with an air speed of 45 m/s (with respect to the air). Additionally, wind blows due SE at 15 m/s.

What is the velocity of the plane relative to the ground (speed and direction)?After 10 seconds how far N/S has it traveled?After 10 seconds how far E/W has it traveled?After 10 seconds what total distance has it traveled?

Do problemspage 71 9, 11, 14a, 16

Independence of Vector Quantities

• Perpendicular vectors can be treatedindependently of each other.

6 s

A boat can move 5 m/s through still waterHow long does it take to cross the water if it heads directly across?

30 m

30 m

A boat can move 5 m/s through still water.If the river flows 2 m/s down stream, how long does it take the boat to cross the river if its motor is aimed directly across.

2 m/s

5 m/s

Water flows

6 s

The boat was still traveling 5 m/s in the x direction

vx= +5 m/s

and it was moving at 2 m/s in the N direction

vy= +2 m/s

But at the same time

Changing the Y-component does not affect the X component

What is velocity of the boat relative to the shore?

2 m/s

5 m/s

2 m/s

5 m/s

R

R = 5.4 m/s

The RESULTANT is how fast the boat is moving with respect to a bystander on the shore!!

30 m

How far did the boat drift down stream

2 m/s

5 m/s

Water flows

6 s

12 m

30 m

What is the resultant displacement?

2 m/s

5 m/s

Water flows

6 s

12 m32 m

30 m

What was its total speed compared to the bystander?

2 m/s

5 m/s

Water flows

6 s

12 m32 m

32 m6.0 s

=5.4 m/s

A boat’s motor propels it at 2.8 m/s with respect to the water.It aims its engines directly across a river which is 68 m across. When the boat reaches the opposite shore it had been carried 24 meters downstream

The time to reach the opposite shore.The velocity of the river.The resultant velocity of the boat. (magnitude and angle)

A boat’s engines can push it at 2.60 m/s relative to the water.If an ocean flows Due NW at 1.1 m/s, Which way should the boat direct its engines in order to go due east?What will its resultant velocity be?

1.1 m/s (ocean current)

1.1 m/s (ocean current)

2.60 m/s(velocity due to engine)

Resultant should point due east

Which way to point the engines velocity to end up going due east?

What is theta?

A= 1.1 m/s

B = 2.60 m/s

45o

Ax= -.78

Ay= +.78By= -.78

Sin() = 0.78/2.6

= 17o S of E

If the resultant has no Y component then the Y components of A & B must cancel.

A= 1.1 m/s

B = 2.60 m/s

45o

Ax= -.78

Ay= +.78

By= -.78

How fast is the ship traveling?

A= 1.1 m/s

B = 2.60 m/s

45o

Ax= -.78

Ay= +.78By= -.78

17o

Bx= +2.5

Is the resultant always the longest side??

NO it is the result of two vectors added, sometimes they cancel partially or wholly.

A

B

Ax

Ay

Bx

BY

Ax Bx

Ay

BY

A current in the ocean is pushing a boat 12 m/s directly Northeast.A wind is blowing the boat 8 m/s 10o South of east.What is the boats direction and speed?

45o

12 m/s

current+

+ east

North

8 m/s

10o

Wind

45o

12 m/s

8 m/s

10o

Current

Wind

Vy

Vx

Vy

Vx

Vy = sin 45 * 12 m/s = 8.5 m/s

VX = cos 45 * 12 m/s= 8.5 m/s

Vy = sin 10 * 8 m/s = 1.4 m/s

VX = cos 10 * 8 m/s = 7.9 m/s

But note direction ( -1.4 m/s)

Add em UP Ytot = 8.5 m/s - 1.4 m/s = 7.1 m/s

Xtot = 8.5 m/s + 7.9 m/s = 16.4 m/s

Vy = 7.1 m/s

Vx = 16.4 m/s

A current in the ocean is pushing a boat 12 m/s directly Northeast.A wind is blowing the boat 8 m/s 10o South of east.What is the boats direction and speed?

7.1 m/s

+

+16.4 m/s

R

R2 = 7.12 + 16.42

R = 17.8 m/s

east

A current in the ocean is pushing a boat 12 m/s directly Northeast.A wind is blowing the boat 8 m/s 10o South of east.What is the boats direction and speed?

7.1 m/s

+

+16.4 m/s

17.8 m/s

tan oppositeadjacent

7.116.4

tan-1 ( 7.1 / 16.4) = 23o

The boat is traveling 23o north of East at 17.8 m/s

east

What would the boat do to use the least amount of power to go directly east?

7.1 m/s

+

+16.4 m/s

17.8 m/s

Direct is engines 7.1 m/s due South

east