Waching daily Jun 28 2017

Hello!

Do you remember this situation from high school physics?

You were supposed to calculate "the work done" by a constant Force (F)

exerted on the car by moving it over a distance s.

At school you learned to take the product of the magnitudes of Force F,

distance s and the cosine of the angle between F and s.

This quantity is often written as F dot s and is called the dot product of Force and Displacement

if you regard Force and Displacement as vectors!

The big dot in the notation explains the name.

In this slide we used vector notation.

That's the reason that we wrote the magnitudes of the force and the displacement

of the previous slide as the lengths of the vectors F and s.

A disadvantage of calculating the dot product in this geometrical way is that you have to know

the angle between the vectors and the lengths.

Often however the components of the vectors are given and then the angle is hard to compute.

But If that's the case, I have good news:

then there is a much easier algebraic way to calculate the dot product!

Let's look how this works.

It is shown for two general three dimensional vectors a and b.

To calculate the dot product you just have to multiply the first components of the two vectors,

then the second ones and at last the third ones.

Adding the results gives the dot product.

With a little practice everyone can do it!

This is our formal definition of the dot product of two three dimensional vectors.

The dot product of two two dimensional vectors is defined in a similar fashion.

Here is a two-dimensional example:

Suppose the force F is the vector with components minus 4 and 3

and suppose you have moved the car over a distance of 3 meters.

Since the displacement is horizontal and in negative direction

you can describe this with a vector with components minus 3 and 0.

The work done is the dot product of the vectors F and s.

With our new formula you can simply calculate this dot product as the sum of the products

of the corresponding components.

In this example the answer is 12.

And now the three-dimensional case.

Suppose two vectors are given.

The dot product is then easily calculated,

again multiply the corresponding components

and add the results.

In this case the answer is minus 13.

In class I would let you think about the question that's on the slide now.

The dot product of a two and a three dimensional vector is asked for.

The answer is that it is not defined.

So remember:

only dot products of vectors with the same dimension can be calculated!

Before connecting the two different ways of calculating the dot product of two vectors,

one with a geometric approach and the other with an algebraic approach,

I will show you some nice properties to do meaningful calculations.

Let's have a look at these properties, valid for vectors of the same dimension.

The first states that the dot product is commutative,

a property well known from ordinary multiplication with numbers.

With numbers you are also used to the second property which combines the dot product with the sum,

better known as the distributive law.

The third property states the connection between dot product and scalar multiplication of vectors.

The fourth property, which is somewhat different, is easy to understand.

Think about it!

It states that the dot product of a vector and itself equals the square of its length.

A property that you are going to use often.

Especially the first three properties make you think that calculations with the dot product

are more or less the same as calculations with numbers.

But be careful!!

Some weird things can occur.

Curious?

Make the exercises!

One of these weird things is the fact that the dot product of three vectors isn't defined.

One reason for this is that the dot product of two vectors is a scalar.

The properties given here can all be checked by using the definition of the dot product.

Some of them are left to you as an exercise.

This example shows how you can use these properties to write the distance between two vectors

as a sum of dot products.

Useful since a geometrical approach is connected with an algebraic approach!

Using the fourth property you first write the square of the distance as a dot product.

Now using the first three properties you can rewrite the result as sum of dot products.

Now you of course want to know what the connection is between the geometric approach

this video is started with and the algebraic approach introduced a little later.

It appears to be a direct consequence of the well known Cosine Law which gives the connection

between the lengths of the three sides of a triangle and one of its angles.

Applying this law on the triangle formed by the common starting point and the two endpoints of two vectors,

you get the result as shown here in terms of lengths of vectors.

You can use the formula derived on the previous slide to rewrite the left hand side in terms of the dot product.

A little rewriting gives the connection searched for.

You may try to do it by yourself in the exercises.

All nice math,

but what else can be done with this concept!

Watch the next video to discover this!

But it's better to practice a little bit first.

For more infomation >> Dot Product - Mathematics for Engineers - Vectors - TU Delft - Duration: 7:49.

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What is a vector? - Mathematics for Engineers - Vectors - TU Delft - Duration: 8:06.

Hello.

Welcome to the mini-course on vectors.

This course is meant for students who plan to win the Nobel prize.

It gives you all the basic knowledge you need to apply vector properties in Mechanics!

If you don't plan to win the Nobel prize, but you want to pass your exam in mechanics,

this course is also exactly what you need.

Take a look at this airplane!

It is surrounded by various arrows.

These arrows have a starting point, a direction and a length.

This picture of a bridge also shows such arrows.

These arrows are not drawn just to give the picture a scientific flavour.

They are very useful for making relevant computations.

In this video, we introduce the basic concepts of these arrows, called vectors in mathematics.

Vector is a latin word, which means carrier.

It was first used by 18th century astronomers investigating planet rotation around the Sun.

A vector is often used in geometry and physics to represent physical quantities

that have both magnitude and direction.

Think of physical quantities like speed, movement, acceleration and force!

Consider this pool table.

If you want to indicate a location on the pool table, this can be done by fixing an origin,

and a horizontal and vertical position of the location, relative to the origin O.

For example: the point (6,4) and the point (-8,2).

If we want to describe a movement on the pool table, this can be done using a vector.

The movement from (6,4) to (-8,2) can be represented by an arrow starting at (6,4) and ending at (-8,2).

If we want to...

If we denote the point (6,4) by A and (-8,2) by B,

the vector in this picture is denoted as AB with a little arrow on top.

There are various operations that can be applied to vectors.

In this video, we consider two of these: scalar multiplication and addition.

We start with addition.

These operations can be viewed from a geometric point of view as well as from an algebraic point of view.

We start with the geometric view.

So consider three points in the plane: A, B and C.

Construct the vectors AB and BC .

The sum of these is then obtained by following the two arrows, first AB and then BC.

This gives you AC .

Now, what to do if the two vectors are not nicely attached,

in the sense that head and tail of the two vectors are not connected?

Well, then it is important to realize that for vectors, only length and direction are important.

In that sense, the two vectors given here are actually equivalent.

When we look at a vector u or v, we have to stress that we mean vectors, rather than numbers.

Unfortunately, there is not just one universally accepted notation to do this.

Often, the u is underlined to stress that it is a vector and not a number.

Other people use u with a little arrow on top and other people use boldface.

In this video, we use the boldface notation.

Remember this is just a convention and don't panic when you see people using other ways

to indicate the difference between a scalar and a vector.

Ok. Now, since two vectors are considered equal whenever their length and direction coincide,

adding two vectors can be done by shifting the second vector such that its tail starts

at the head of the first vector.

In this picture, first shift the vector v such that its starting point is exactly the endpoint of u

and then add the vectors as before .

In practice however, if you want to add two vectors, usually you first shift their starting points to the origin.

Then the sum of the two vectors can be obtained by using the so called parallelogram law.

Vectors can also be multiplied by a number, often called scalar in this context.

Briefly, multiplying a vector by 3, means its direction is left unchanged

and its length is multiplied by this number 3.

For instance, consider this vector v.

3 times v is given by this.

Multiplying v by minus 3, on the other hand, will give a vector which has opposite direction to v,

and its length is still 3 times that of v.

Ok, let us now look at vector addition and scalar multiplication of vectors

from an algebraic point of view.

The standard algebraic representation of a vector in the two dimensional space,

is by its end point, assuming the vector starts at the origin.

The coordinates of the end point are stacked on top of each other

and collected between two tall brackets.

For instance (4,1) and (-3,5).

Using this representation, addition of two vectors is quite natural.

The numbers 4 and 1 are called components of the vector (4,1).

Geometrically, we see that u plus v starts at the origin and ends in (1,6) .

The representation of this sum vector is (1,6).

Note that the first component 1 of the sum vector u+v is simply the sum of the first components

4 and minus 3 of the vectors u and v.

Similarly, the second component 6 of the sum vector u+v is just the sum of the second components

1 and 5 of the vectors u and v.

Multiplying the vector u equals (4,1) by 2 gives (8,2), of course.

Multiplying u by minus a half, leads to (-2,-1/2).

This can be seen from the geometrical representation as well as from the algebraic representation.

The examples up till now were given in two dimensions.

In three dimensions, you can take exactly the same approach.

A vector has a direction and a length.

It can be represented by an arrow.

Here you see the vector u with end point (1, 2, 3).

Adding the vector (1, 2, 1), leads to the vector (2, 4, 4).

This can be seen by moving the second vector's tail to the first vector's head

and travelling along the connected arrows.

Alternatively, the sum of u and v can easily be obtained by adding just the components of the vectors.

In this video we defined vectors in two and three dimensional space

both geometrically by magnitude and direction, and algebraically by components.

You also learned how to apply the following operations:

addition of two vectors and multiplication of a vector with a scalar.

In the next video you will learn more about the length of a vector and distances between points.

Stay tuned!

For more infomation >> What is a vector? - Mathematics for Engineers - Vectors - TU Delft - Duration: 8:06.

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Length of a vector - Mathematics for Engineers - Vectors - TU Delft - Duration: 5:20.

Hi there!

How strong is a force?

What is the speed of an airplane?

These questions can be answered numerically.

More precise answers can be given using vectors, as introduced in the previous video.

The direction of the vector indicates the direction of the force.

The length of such a vector gives the magnitude of the force.

The length of a vector can be computed using the Pythagorean theorem.

This theorem is named after Pythagoras of Samos.

This mathematician and philosopher lived about 500 years b.C.

and it might be nice to just gather some information on this person.

Let's not do that here and focus on his theorem.

Writing the vector u in two dimensional space with components u1 and u2,

the length of u is given by the square root of the sum of u1 squared and u2 squared.

The length of a vector, also called its norm, is denoted by double absolute values.

In three dimensions, the theorem is similar, and the length of u is given by the square root

of the sum of squares of the three components, u1, u2 and u3.

Unit vectors are vectors with length one.

All nonzero vectors can be decomposed in a unit vector determining its direction,

and its length.

Let's see how this works:

Consider the vector (2,0).

It can be viewed as the unit vector (1,0) multiplied by the scalar 2.

Right?

Now this is what you did to get this remarkable insight:

you just divided the original vector by its length!

Consider the vector (3,4).

The length of this vector is the square root of 9 plus 16 which is 5.

So the unit vector with the same direction as (3,4) is the vector (⅗ , ⅘ ).

Just like before, the vector (3,4) can be seen as the scalar product of its length

and its associated unit vector.

In general any nonzero vector can be decomposed in this way.

If you think of a nonzero vector in two dimensions yourself,

then its associated unit vector will have its end point somewhere on what is called the unit circle.

In three dimensions, the idea is the same: the vector (3, 1, 2) has length square of 14,

so it can be written as the scalar product of the square root of 14, its length, and...

..the unit vector which has the same direction.

The general statement is again that any nonzero vector is its length times the unit vector

with the same direction.

Among all unit vectors in three dimensional space, there are three special vectors:

the standard basis vectors.

These are the vectors having all but one component equal to zero and one component equal to one.

i, j and k constitute a so called basis of three dimensional space.

Every vector in this space can be written uniquely as a linear combination of these

three basis vectors.

Indeed, considering the vector (2, 2, 3) as an example, there is exactly one way to write it

as a linear combination of i, j and k.

Besides the length of a vector, one can also define the distance between two vectors u and v.

It is defined as the norm of their difference u minus v.

This distance is exactly the same as the Euclidean distance between the two end points.

As an example, consider the vectors (1,2) and (-1,3).

The vector u minus v is then given by (2,-1), which has length square root of 5.

This leads to the statement that the distance between (1,2) and (-1,3) is square root of 5.

In two dimensions, we saw that all unit vectors end on the unit circle.

Where are the unit vectors in three dimensional space?

So, where can the vectors with length one be found?

You won't be surprised that the answer is given by the unit sphere.

A related question, one could ask is which vectors are at distance 2 from the vector v=(1,2,3)?

Those are the vectors u for which the norm of the difference vector u minus v equals 2.

And geometrically, these are the points on the sphere centered at v, with radius 2.

Now you know about the length of vectors, unit vectors and distance between vectors,

the next video will introduce a related concept that is very important in many applications:

the dot product.

You know about scalar multiplication of a vector.

The dot product gives meaning to the product of two vectors!

Curious?

Don't hesitate and watch the next video!

For more infomation >> Length of a vector - Mathematics for Engineers - Vectors - TU Delft - Duration: 5:20.

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What is Orthogonal Projection? - Mathematics for Engineers - Vectors - TU Delft - Duration: 1:45.

Hi there!

Let's have another look at the moving car.

If you want to compute the force that causes the car to move forward, here is what you need to do.

Suppose the car is dragged along by a constant force F under an angle alpha.

The force F can be decomposed into two perpendicular forces:

a vertical force lifting the car,

and a horizontal one causing the car to move.

This horizontal force (minus the friction) is the force that actually makes the car move forward,

so we need to calculate the blue vector!

In a general setting the situation is as follows:

Given two vectors u and v, you want to find vectors u1 and u2,

such that u1 has the same direction as v,

u2 and v are orthogonal

and the sum of the blue vector and the yellow vector equals the green vector u.

The vector u1 is called the orthogonal projection of u onto v.

There is even a separate notation for it.

Imagine light shining from above onto u then u1 is in fact the shadow of u on v.

In the next video you will learn how to actually calculate orthogonal projections.

You should now do the following exercises to internalize the meaning of the concept just introduced.

This is really needed to understand the calculations done in the next video.

For more infomation >> What is Orthogonal Projection? - Mathematics for Engineers - Vectors - TU Delft - Duration: 1:45.

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Circuit boards used for prototyping - SGW Designworks - Duration: 3:14.

hi my name is Ken Oakeson and I design and develop embedded systems for SGW designworks

and today I'd like to talk about some of the boards that you might

find used in the development process of a printed circuit board for an embedded

device or the Internet of Things so you may have heard of the Arduino and

Raspberry Pi devices they're used by a lot of people for instructable type

projects and they're also good for prototyping we like them because they're

inexpensive they come with lots of accessories that can snap on there

called shields those come in different sizes with

different capabilities like sensors or lights or connectors and then the the PI

is a full-fledged Linux computer this is the Raspberry Pi 3 has all the

connectors for Ethernet HDMI you could use it like a regular computer and so it

also has powerful libraries that are available to it for things like computer

vision things like that there's also the smaller Raspberry Pi 0 it's a wireless

computer much smaller form factor so it gives us a lot of options for

prototyping something but we typically don't use these in the end product

because they either cost too much relative to a custom printed circuit

board or they're not available in the quantities that we might need or the

size of them is too large so when we get further along in the development process

we like to use a custom printed circuit board and in the end that's going to

have exactly what we want on it and nothing else nothing extra and we use

the lowest price components that we can and then connect any sensors or devices

that we need to so to get to this a lot of times we'll use an evaluation board

from a vendor like microchip in this case or Texas Instruments and the

evaluation boards typically have a the chip on them the microprocessor and

perhaps a Wi-Fi or Bluetooth connection this particular one has Ethernet on it

and then it may also have a few cents SURS or a few buttons or LEDs for

testing things out and then a socket to allow you to connect things like we

talked about for the raspberry or for the Arduino on the Raspberry Pi but we

can we can wire in the actual sensors and controllers that we're going to use

see if they work correctly once we feel comfortable with that then we can move

to the custom board and that's something that the the programming on this is a

little more involved we call a bare-metal programming it's a little bit

lower level than what we'd be doing on these devices but it gives us the

minimum code size again so that the chip and the memory can be smaller and

cheaper and that can take us all the way to the manufacturing prototype so that's it

in a nutshell thank you

For more infomation >> Circuit boards used for prototyping - SGW Designworks - Duration: 3:14.

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Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers - Duration: 11:08.

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers

For more infomation >> Colors Horse Videos For Children Horse Finger Family Rhymes Color Horse Cartoons For Toddlers - Duration: 11:08.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:18.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:18.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 1:29.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 1:29.

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Jill Kassidy Q&A for FA - Duration: 1:20.

- [Narrator] Fucking Awesome.

(typing)

- I had so much fun shooting for fuckingawesome.com today.

(typing)

I've never had the chance to see 50 Shades of Grey,

but I've always wanted to so it was fun getting to do

that sort of thing.

(typing)

I don't really have crazy sex off-camera.

I like more passionate stuff so yeah.

(typing)

If I could only eat one food for the rest of my life

it would be Mexican food 'cause that's my favorite.

(typing)

I guess the most interesting thing whenever I meet a fan

was whenever one of them paid me to lick my toes

and take pictures.

- [Narrator] Fucking awesome.

For more infomation >> Jill Kassidy Q&A for FA - Duration: 1:20.

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Dinosaurs Surprise Eggs For Children 3D Giant Dinosaur Surprise Egg Hatching #Dinosaur #surpriseggs - Duration: 59:04.

Dinosaurs Surprise Eggs For Children 3D Dinosaur Giant Surprise Egg For Kids #Dinosaursurpriseggs

For more infomation >> Dinosaurs Surprise Eggs For Children 3D Giant Dinosaur Surprise Egg Hatching #Dinosaur #surpriseggs - Duration: 59:04.

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All eyes on FC Cincinnati for Wednesday's game - Duration: 2:39.

TONIGHT IS THE NIGHT F-C

CINCINNATI FANS HAVE BEEN

WAITING FOR. THE OPEN CUP

MATCHUP WITH THE CHICAGO FIRE

IS ONCE AGAIN PUTTING F-C

CINCINNATI IN THE NATIONAL

SPOTLIGHT.

THAT'S WHERE THEY'LL HAVE TO

BE IF THEY WANT MAJOR LEAGUE

SOCCER TO CONSIDER ADDING THE

TEAM FOR THE 20-20 SEASON.

NINE ON YOUR SIDE'S ALLY

KRAEMER IS LIVE INSIDE NIPPERT

STADIUM -- ALLY HOW MANY FANS

ARE THEY EXPECTINGTONIGHT?

MORE THAN 30-THOUSAND

SCREAMING SOCCER FANS ARE

EXPECTED TO FILL NIPPERT

STADIUM - THEY SET A RECORD IN

FIRST OPEN CUP GAME WHEN THEY

BEAT THE COLUMBUS CREW - NOW

THEY'RE FACING ONE OF THE BEST

TEAMS IN THE M-L-S.

ESPN ANALYSTS WATCHED PRACTICE

TUESDAY AND TELL US THEY'RE

ANXIOUS TO SEE HOW SOCCER HAS

TAKEN HOLD IN CINCINNATI.THE

VOICE OF ESPN SOCCER - ADRIAN

HEALEY - SAYS THE ATMOSPHERE

HE SAW AT THE FIRST OPEN CUP

GAME WAS AS GOOD AS ANYTHING

HE'S EVER SEEN ANYWHERE IN THE

M-L-S. IT STILL SURPRISES SOME

FANS - THEY EXPECTED ABOUT 10-

THOUSAND FANS - NOW THEY'VE

TRIPLED THAT.DIE HARD FANS AND

FIRST TIMERS ALIKE WILL MARCH

INTO NIPPERT STADIUM - FROM

THE PRE-GAME PARTY AT

MECKLENBERG GARDENS. ONE OF

THE LEADERS TELLS US HOW

THEY'RE MAKING CINCINNATI

STAND OUT TO THE M-L-S.

034311 Ronny Salerno / Board

Member, Die Innenstadt You go

to a baseball game in

everyone's just walking around

and not paying attention. You

come to an FC Cincinnati match

or any soccer match everyone

is fixated on the action on

the pitch everyone's really

into it the chance in the

chairs they react to what is

happening on the field people

are just paying more attention

and they're really excited

about it while it's going on"

FANS ARE READY - IF IT'S YOUR

FIRST TIME COMING DOWN FOR A

GAME - COME EARLY AND EXPECT

HEAVY TRAFFIC AROUND THE

STADIUM. COMING UP AT 6:00 -

HOW TO GET TICKETS FOR WHAT IS

ARGUABLY THE BIGGEST GAME YET.

REPORTING LIVE, AK, 9OYS.

ESPN TWO HAS THE GAME COVERAGE

AT EIGHT P-M TONIGHT...BUT

BEFORE THEY GET STARTED...

JOIN US AT NINE ON YOUR SIDE

FOR THE PREGAME SHOW.IT'S

CALLED "QUEST FOR THE CUP"...

THE 9 ON YOUR SIDE SPORTS TEAM

WILL HAVE LIVE COVERAGE AT

NIPPERT STADIUM... STARTING AT

For more infomation >> All eyes on FC Cincinnati for Wednesday's game - Duration: 2:39.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:15.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:15.

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Dem & GOP Governors Hickenlooper and Kasich Team Up To Urge For Bi-Partisan Solution To Healthcare - Duration: 5:00.

For more infomation >> Dem & GOP Governors Hickenlooper and Kasich Team Up To Urge For Bi-Partisan Solution To Healthcare - Duration: 5:00.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:28.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:28.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 3:26.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 3:26.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:22.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 5:22.

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Cow Vs Dinosaurs Finger Family Rhymes Dinosaurs Movies For Kids Elephant Rhymes For Children - Duration: 17:50.

Cow Vs Dinosaurs Finger Family Rhymes Dinosaurs Movies For Kids Elephant Rhymes For Children

For more infomation >> Cow Vs Dinosaurs Finger Family Rhymes Dinosaurs Movies For Kids Elephant Rhymes For Children - Duration: 17:50.

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Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 4:31.

For more infomation >> Local Electoral (Equitable Process for Establishing Māori Wards and Māori Constituencies) Amend.. - Duration: 4:31.

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Games for Kids Learn Colors with Subway Surfers Mexico City Gameplay Video iGame Kids Cartoons - Duration: 10:23.

For more infomation >> Games for Kids Learn Colors with Subway Surfers Mexico City Gameplay Video iGame Kids Cartoons - Duration: 10:23.

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Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes - Duration: 15:20.

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes

Colors Dinosaurs Cartoons For Kids Gorilla Vs Bear Real Fights Pig Finger Family Nursery Rhymes