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Hi there.

Obtained in the previous session

in three dimensions on which we generalize examples will do.

You'll recall the initial plane We saw vectors.

We have identified four related transactions.

Addition, multiplication by a number of internal and internal as multiplication and vector multiplication.

This process in the previous session and We generalize the concept to three dimensions.

Here, the same arithmetic operations as well as a We have seen that there are triple vector product.

Despite being one of two types of did not dwell on.

But by multiplying a third complex vector We saw volume can be obtained.

Now we will make examples of them.

How the process of the usual We will learn to place

With these concepts and examples will solidify.

Following our example.

Us given four points A, B, C, D points.

As you can note a simple point we chose

accounts so that they do not interfere unnecessarily.

Because essential to understand the concept of No need for complex calculations.

No matter how simple account of sophistication We can not return to the core.

Our questions are:a vector B. A'yl combining to define the EU.

C A'yl to define vector v combining

and we define the vector D in A'yl combining.

Now to find these vectors, and K, We want to find their size.

Ie the angle BAC to point B in the middle the

Gradually the points A and C formed We want to find the angle.

A, B, C points obtained by combining We want to find the area of a triangle.

Again, this A, B and C of the three-point We know that forming a plane.

Perpendicular of the plane defined by these three points We want to find vectors.

We also want to find even a unit vector.

Of point C, B and D is a plane We want to find the distance.

Gene A, B, C and D, see points

Let me show you my hand, A, B, C, D point though three vectors are created here.

Here again, AB, AC, AD vectors We gave.

It consists of three vectors of hexahedron We want to find the volume.

Pulling them in parallel, where a this side of the base line parallel surfaces

surfaces by obtaining a crystal-like We want to find the volume of a structure.

Gene A, B, C are points.

A, B, C, to get them to that point, we combine

When that occurs because of a right triangle, B,

and one point C on the x axis, one

on the y axis, an ideal Z axis over it.

This will connect the dots.

He's also a starting point for combining zero zero

a pyramid in zero coordinates, other We will obtain the pyramid.

We want to find the volume of this steep pyramid.

Also find this kind of a triple product we want.

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First, the location of this point by drawing a little understand the geometry, let's understand.

You will recall again in three dimensions we have two We try to show size.

As a reg, but in a region of the painter

from around the world on a global level with the projection

A perspective that maps appearance

x, y and z axes orthogonal axes but we

With such a perspective representation of necessity

We show the plane.

Components of point zero zero one.

So, just a direction x in the x direction

component there, second only to point B There component.

Therefore, only has component in the y direction.

Thus is here.

is on the y-axis.

Similarly, point C only zero zero x and y components.

z is the z-axis component of the three over it.

See the interview here than just a pyramid formed.

One point to center point By combining, also this

In addition to the three points of a point D given, it is hardly so simple

coordinates, but have not not two, three, four, given a point.

We start from the simplest.

Finding the components of the vector AB We want the EU vector

wherein the components of this vector, see We want to find.

You'll recall that combines two points To find Vetori

I'll start from the coordinates of the second We will first remove the coordinates.

This first of it from the first second like taking income.

On the contrary, the first of the second

From the coordinates of B's coordinates will bring.

So two zero zero one zero We will remove zero.

We have an interest minus one from scratch.

We have two two zero interest.

We remove the zero-zero-zero.

That means that the components of the vector AB.

AC vector're looking at.

Yet again the second point of the AC vector We will first remove from the coordinates.

So it brought out a negative one from scratch.

We remove the zero-zero-zero.

We remove three zeros from the three.

Similarly, we also want to find AD.

The first gene in the name of the second components, we remove one of the two.

We remove three zeros from the three.

Remove the four have zero to four.

That AD vector.

Find their equally sized easy.

Because the mean length of the squares of the components We will gather to take.

One plus four plus zero.

One plus zero plus nine.

One plus nine plus 16.

We find that the lengths vectors.

One of the issues to be considered here one

The second point of the components of the first extract the coordinates of the point.

Now that the angle BAC, we want to find B, A, C

i.e. the middle point of the angle We want to find.

Now that we can find two types of terms.

Finding the cosine or sine.

When we look at the mean cosine BA BAC AC

EU AC vector with mean vector The angle between.

It is located to the inner product.

We found the EU.

AC also were found.

So minus the bier from the multiplication of the two Once a, a,

twice, three times zero is zero out denominator, the denominator.

Have the length in the denominator.

Five EU root length, root length AC,

10 wherein the control gene is We can.

One plus nine, 10.

Here, too, a plus-four-five.

Square root thereof.

A divided by the square root of 50 turns.

This small calculator in your hand from the Or from a table

If you look at this one because it was close to 90 degrees a fairly small number.

If it turns 82 degrees.

We can find the same size from sinus.

You will remember sinus inner, inner product There was not the vector product.

As with the above-like product but Instead of inner product vector product is coming.

In the denominator of the vectors in the same way have lengths.

And before switching to a three-vector multiplication times

easier to do with the interior because the cross number of turns.

Although the vector by multiplying though we provide We can look at.

In fact, we will do it for us already here asked the A, B, C points

See formed following A, B, C points located in the area of triangle appears.

Mean vector multiplication means the area.

Vector multiplication mean to say determinants.

Which determinants?

i, j, k we will write as first line.

The second line of the first vector components minus one

and AC minus one zero three two zero.

We will write them open.

When he opened i, j, k, we can do without writing need.

The first component of the vege i, the row and the column will take.

Only two zero zero three remain.

So it turns out six here.

I'll start with the second in the negative.

j the column and row are taking.

There is a minus minus three.

Was plus three.

The last column of the first line and for k We are increasing.

Zero minus two but less than one bi because it is plus two.

This means that the vector of the triangle area, the half of the length of the vector multiplication.

Where b is two more lines of I had to.

When we look at this

The length of the frame 36 of gold.

Three of the nine frames.

Two of the four frames.

45, 49.

49 is the square root of seven.

Seven divided by two turns.

As the product of these vectors provide here for six

The length of three to two the length of AB and AC dividing.

EUR has now been seven long above this The product.

B and AC in the denominator of the length of the root 50 We found he gives.

Seven out of 50 are divided.

Is that correct?

Yes.

Because the sine squared plus cosine square I need to.

A sine-squared divided by 50.

You divide 50 a square cosine sorry.

50 sine-squared divided by 49.

That is a plus 49, 50 divided by 51.

So it seems to provide.

Unit vector perpendicular to the mean EU

AC generated by the normal vector to vector We want to find.

E that e, is obtained by multiplying the vector.

A B C in the plane defined by these three points AB and AC can get.

They have found the vector product.

Six, three, two were.

If we divide this longitudinal length took seven involved as you can see.

Here we find a unit vector.

Finding the point D is the distance from this plane we want.

Now here but I did not try to draw schematic

If you look at the plane of point D apart from.

Let us unite it at any point.

A point.

To the point when we combine AD vector involved.

This plane is a vertical vector.

Unit vector perpendicular.

We found here.

These two e, we take the inner product you can see

The distance D from the plane as interests.

So the AD on the projection.

At the same time D to be the base.

Here because of this it together with n parallel.

Here is a complete rectangle we have created for.

This inner product is very easy.

Here we found the vector.

We found a before AD vector.

You turn he was a three or four.

From the coordinates of D's in After I remove the coordinates.

Get it by multiplying the above six plus nine plus eight

i.e., the first component of the first component, with the second component of the second, third third

try as much as six to nine fifteen.

Eight more involved twenty-three denominator.

There is already a seven in the denominator.

Now we are here, but we combine A'yl C'yl de

we do the same process again to combine the same things took off.

Suppose a point here because the CA.

E, that the perpendicular vector of point C the same.

Which dot it for him

If you combine the same projection plane you will find.

You can find the same height.

Here as a test, providing

Let us take as example a CD or BD Let's take.

So combine with n perpendicular to B, D, internal multiplication or

D. C, combining with n inner product We need to find when you get the same results.

Because of this we will assume in the plane here here B C D like that here.

Alırs which point, if you combine combined, even what they other one

Even the combined perpendicular vector to the point We are looking for on the projection.

At this point all of orthogonal vectors e the same.

And there's also still the same as providing We see the results that we found.

The volume of the hex-shaped parallelepiped I want to find.

This volume with this trio that the cross 've seen.

We found the EU.

We got AC.

In the case of AD have found.

This can be done in several ways.

The most practical course of these determinants to calculate.

First line u, v, w said.

AB, AC and AD, respectively.

This component to the first component of the EU.

The AC component, wherein a negative zero, three.

AD components of one, three, four.

When we hit it, this determinant to

When we calculate the triple product to 23 involved.

If you can work really well.

According to the first line of zero we get Let's think.

See a minus times minus the zero It was nine plus nine.

We'll get minus two.

Less than two thirds minus four to minus, There are seven minus 14 to minus here.

This zero.

So the first one was minus nine.

C to E, but minus the merger is multiplied plus nine.

Here minus minus fourteen, but by two, Because the gene plus 14 multiplied by the axis.

14 plus nine still doing 23.

Another way of the hex-shaped body base and the height of the bump.

Base floor area multiplied by the EU and the AC giving.

A stylish, it had calculated.

You turn has six three two vector.

There was also a three or four AD.

This type of three-four four We see that come immediately.

A third way that all of these genes comes to the same place.

We found the area of triangle ABC.

Floor area that is twice that.

Seven divided by twice the area of triangle ABC We have found two.

The point D to the surface of the triangle We found the distance.

We found him 23 divided by seven.

That 23 out of them gets hit again we see.

But of course the most practical of this determinatl to calculate.

Too much thinking, thinking in geometry not necessary.

See if he thought the volume of the pyramid pyramid occurs here again, occurs.

By volume of the base area of the pyramid height is one-third hit.

This is something that is known from geometry.

Let's get right to me?

One of these two vectors by the triangle,

Half of the vector cross product of the OB.

Therefore, a divided three times a slash We will multiply the two.

There comes a divided six.

Plus CE, and its height is multiplied.

This is a steep pyramid of height height to clear here.

Height three.

So you can get out of here.

The area has been divided into two triangles 27.

If this treble 81 divided by three.

If we divide that a six turns.

Here again, this is from the determinants If we write the determinant of a zero-zero OR.

And two zero zero zero zero three.

It will divide the six.

This six day when you open the determinant.

According to the first row once, let's open a.

Once these two duality.

Two times three.

Secondly zero.

Drops.

Zero third drops.

There's only six remain.

Comes from a six divided by six.

I leave you with this triple product at work.

You can make two kinds.

We were hit by the EU and AC, were found.

AD know that.

These vectors multiplied by the interest taken or b there were also formulated.

It also turns out that the implementation of the formula.

We want to give you an assignment.

Point a little more different.

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Yet given four points.

A fifth of a given point.

Here you have an unknown alpha.

Gene on the lengths of these vectors to find I want.

E, to find the angle of the triangle

area, all of these as the perpendicular vector We want to find.

Lastly, in Bi, B, C, D

For points not on the same plane I want to find alpha.

For this, an AB, AC, AD

vectors, triple product of this volume was giving.

This volume is zero nor an alpha is optional?

They strongly recommended that you do I would.

Because unless you yourself just me

Follow my solution is completely Your pekiştiremezs works.

I've done my solutions Watching quite a few

You are learning, but a little bit of yourself I need to do.

This is also a sport you like when making tennis players

Or how successful basketball players is going to watch on TV is not enough.

You need to go to the field to play yourself.

I would strongly advise that respect.

Now here we finish this issue again.

We e, making operations with vectors We learned.

We learned the meaning of this process.

Now verify these processes in space and planes will apply.

This is important for the following.

Verifies the simple vector functions in space.

In the plane, planes, the most simple two-

variable function.

So we en vector functions as follows, en, have defined.

Even a single independent variable, three, compound.

Full of numerical functions of two variables vice versa.

One independent variable dependent variable bi two.

If the most basic of these functions is not linear.

Thus they themselves significant What is this confirmed and planes.

Both are now progressing steadily, it next

chapter in space and space curves We will examine surfaces.

They are also the most simple lines and planes because

them to stand in the natural cut-off point, 're standing.

The next session of this concept, of these lines and planes

of how to build the vector By applying what we have learned we'll see.

Both preparations will have made bi vector function of two variables, and

Gain access to the surface of the function the.

Goodbye.