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Hi there.

In the previous lesson vectors 've seen.

We started before the plane in space We have seen cases of vectors.

Of the most basic arithmetic operations with vectors We have defined.

Collection, with a number of bi multiplication, inner product and vector multiplication.

When we came to this dual vector space multiplication as well

as well as the triple vector product We have seen that coming.

The meaning of the inner product of vectors multiplying a projection vector

product and triple product in an area We have seen a volume.

Them rather as tools but many of them have developed your

in physics courses and other courses as well as the You already kullanıyos vectors.

Forces shows.

Shows the electric field.

B shows the current.

Now that we have learned

vector operations on lines and planes We will apply.

They have value in two senses.

BI them important in their own right.

Secondly, in most cases local curves Thank represented by a true.

We are closer to the true one.

For example, in the plane tangent to the curve at the quite a few of us actually carry information b, gives.

Similar to the space of curves on the simple structure is confirmed.

Thus, with the right following them we shall see in section

an important preparation for space curves will be.

After the two planes in a similar way we will see

section of the overall surface in space We will examine.

Planes also their most basic form.

How is it that by drawing a tangent to a line a near point about

If you are obtaining information about the curve, but important We're learning more about a curve.

Now also a tangent plane to the surface, plane passing

To understand the operations that area near The most important tool.

Therefore, they are, in themselves a purpose, but more complicated

a tool for curves and surfaces.

This is just what I said it a little more No math,

If the only independent representation summer variable but

If the dependent variable vector b 're getting a space curve.

Why?

Because univariate geometric objects gives a curve.

There are only one degree of freedom.

Because that barely moves on the curve b You can scroll on the curve.

If you are out in two directions perpendicular to the curve Are you separated from the curve.

You can stay on the curve for her.

Similar thinking two independent variables

but that is a number of dependent variables

shows a surface in the space of functions.

Shows why the surface?

With two degrees of freedom because is the surface geometric objects.

On such a surface in both directions You can scroll.

But in the direction perpendicular to the surface to move If you try, so

that the degree of freedom to use the

If you do not want to stay on the surface of the separated can be.

A second function so that Shows surface.

The simple structure of these functions to t wherein is a linear function based.

That is a constant vector x vector b plus

the variable t is constant vector b multiplying.

Similar in mind that most of our functions a simple structure

the first force plus the x and y are constants the first force.

These are the simplest types of functions.

Now, it has been understood as a geometric then

equations of these objects geo equations How to remove?

B both have a common structure.

Both receive a point in space.

You can also choose a direction.

To find the equation of a line that from the point

All points in this direction that the direction of belongs to the right.

Similarly, choosing a point in space and

At this point it straightened b You choose.

All points on the surface perpendicular to this direction, creates.

This surface is a plane.

Wherein all points the same In line with a line that is true.

Once you understand it geometrically remove the supremely simple equation.

Want to pin point accurate fixed at zero Want Whether a fixed vector surface.

x is zero, y is zero, z is zero constants is defined.

In addition to this, in both cases a We choose the point on moving.

p point here moving in here.

Is moving along strike or to n is replaced by staying upright p.

However as the variables x, y, z are We koyuveriy.

Fixed line slightly different impressions

Consider the matter involved as to each other.

Let u and u to w'yl.

of n, A, B, C, denote by.

Why are you doing this?

Because of these issues internationally This notation is used in writing.

Those who want to look at the international literature these

difficulties in notation, this notation bi so that they can draw.

Now that we have the right to choose p is zero.

We chose p.

This creates a vector has both.

This vector, the vector in this geometry algebra structure on

coordinates of these points come from.

But there, in the language of geometry before crossing We call this vector following.

p is zero in the direction of the vector p.

Thus in a multiple thereof.

Variable that tells how many times that it is t parameter.

Or t variable, t argument.

In the plane of the same idea again we say.

This vector is perpendicular to p n p is zero.

Others learned to be.

Other means to the inner product is zero means.

So it's right in the language of geometry p is zero equation

once u p t, plane geometry, language p is zero also shown

wherein p is zero, with n other product.

Now in terms of their components Let's write.

Yet here the same way your eyes you might find you're getting into.

the second point's coordinates of x, p, p is zero

x coordinate of the first point of zero is to.

So p p x minus x is zero becomes zero.

once u t.

If we write it in terms of components, there first component x, x is equal to x is zero.

The first component of the u.

Similarly, the second component and the third compound.

In the same thought, the equation of the plane, this Let obtained as algebraic time.

We knew it from the geometry.

n the inner product of p is zero p'yl zero.

If we write this in terms of coordinates p zero p

same here as x minus x is zero is a vector.

So that the inner product of the vector with

vector in the plane perpendicular vector zero we obtain.

This same operation denominated coordinates

If we write here the first coordinates x and terms.

Secondary terms of y.

The third also in terms of z.

for the A, B, C he said.

This meant that the first to take the inner product, coordinate among themselves

second coordinate among themselves third

coordinates with each other to multiply means.

As you can see we make this multiplication to the Times x plus

B times C times y plus z on the left side remains.

Of times x is zero, y is zero B times, C

Once z is zero minus in front of them We are bringing the right side is checked.

Because this is that the number of bi D we say.

D is made in this way in the same job.

So the equation of a plane

x and y and z

with only their first force a linear equation.

Similarly, the line equation at x, y, z, and also

t is the first that is linearly forces and their equations.

This supremely simple and me, this logic also extremely important to tell.

Right relationship with the bi-plane rule.

Because always.

Almost all of us from middle school We know the correct plane.

We have the right to the plane of said diol.

y coordinate of the fixed point When removing, of x

NEW remove, coordinate divided by fixed point here we obtain the slope of this line.

I have given the slope of the vector E of this ratio of components.

This time we organized a y equals x was happening, plus b.

It as equivalent as we can.

If we account that t here.

t of x divided by x minus x divided by zero.

Edit your place, put here by the same we obtain the equation.

So here on the old information in the plane To see the equivalence with the right.

In space no longer in the equation e, b one No slope.

Because of the slope of three, in different planes have values.

Here to proceed to a more algebraic easy.

If we allocate t t we solve here such as e, x minus x zero

u section, section V of y minus y is zero, to be the z minus z

towards zero as part of the SRSG, We exclude from the equation.

Here, though, as it seems three equations There are also two independent equation.

Because synchronize with the first second of If you synchronize with the first third example,

directly with the second third of the equation the right to self-provide.

When you edit each of them See the merger of the two together, let b.

t we forget now.

Because all the process for separating t 've done.

x minus x minus y y Reset section zero.

So it between X, y linearly relationship.

We saw a little before e e, x,

y, z, a linear correlation between The data plane.

It's a plane.

Where z does not appear, but no harm.

Well, because the correct plane could also appear in the equation of.

Then B is a horizontal line y is be brought.

B y is equal to the fixed value of all point of the geometric

Here, z is the place to be do not mind the look.

When we get to the first third equation here

y does not appear, but still do not mind.

This is a linear X, z of relationship.

This plane b.

Each of which means a plane show.

We are looking toward these two planes intersection.

E as we know it geometrically.

Thus to a plane, of a line to We can show two kinds.

Three components in terms of a parameter t expression

or by providing the intersection of two planes.

Here the plane equation x plus B y We said our equals E plus C.

Here, four types of frequently encountered There are private planes.

If you say x is equal to one of constants, x It is equally hard to say that all

x is equal to a fixed point on the to have, which means that in the plane.

M x is equal to that passing through the vertical is a plane.

Similarly if Y is N y is the

N y coordinates of all points at a fixed It is the plane.

Y in the vertical plane of this plane.

Similarly, we say that z is equal to Q the whole point Q

the x-y plane is within a shows the horizontal plane.

We will encounter them frequently.

Similar equations without the intermediate compound When we think of z are independent.

So we know that x and y in the plane of a It is the right of the equation.

But it's true all the points on the provides this equation.

But the intermediate compound is unclear.

EUR restrictions on our already here for lack of them, that this

overall slightly better than two vertical shows the plane.

Now the equivalent of lines and planes Let's look at the definition.

We know that if we choose two points of these two passes through the point of a.

So this is a point and

we have identified the right direction carries an equivalent equation.

This second bi identification, identification way.

While we were on the plane to a point and this point other

We define the vector of the plane, but such As

defines a plane intersecting the two right.

Also, the plane passing through three points is defined.

Sometimes we look at the tables in our homes.

Mas for that four-legged, feet between

If there is little difference table flat, does not remain constant.

That also shows that three of

You can not receive more than bi-plane at the point to identify.

Because each of these three time, four-point bi can define the plane.

Bi plane may be outside of them.

There are infinitely many points in the plane, of course,

but above all restrictions is there.

Because that's all orthogonal vectors E, other consists of the right.

Now let's see examples.

This is so, it is not difficult these concepts concretely.

Given a point.

One two three point's coordinates.

It passes through this point of the vector b given.

Especially for the sake of simple calculations very We chose the simple figures.

Our first question is defined in this way find the correct equation.

Secondly, given us two points.

C and D points.

I wonder if it's the right spots on the is not it?

So the correct way.

Point C on this line may be or may be outside.

D. In a similar manner also.

How will you check it?

The correct equation is as follows found.

variable x, x is zero hard, hard.

X x is zero vector means, wherein We were given zero vector x.

One is made up of two or three.

once u t.

we give u the convenience of a one Whether he.

So as you can see us x but it also gives vector vector x

i.e. as a function of a t a vector function.

If we divide it with components x a plus t, y two

plus t, z be three plus t.

I wonder if this is it on the right spots?

How can we check?

This inspection as well.

this x

Type C y z place it here.

C for three separate insert te we obtain the equation.

See where we saw the first one t, we find the two.

In the second, we find zero t.

We find t minus three in the third.

So these values contradict each other.

Therefore, the right of point C does not satisfy the equation.

Therefore, this right is not over.

When we look at point D it coordinates given as three four five.

We are writing to the coordinates of the three D's.

We are writing the y coordinate of four.

yaji z coordinate, we are writing five.

This is an unknown in the equation, all three of t There are three equations.

T see the two we find here.

Here we find two t's again.

Here we find two t's again.

So the same results from each of the three equations We found a consistent result of this.

Therefore t right of point D

and therefore the correct equation gives above.

Now our example here, with true bi I am finalizing.

Thereafter görecez different examples.

We will see examples about plane.