Find the area of the parallelogram as a given vector. Vector vitvir vector. Zmishane TV vector. Rozrahunok of the dozhin of the sides of the figure, given by the coordinates

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Think back, what is a vector TV.

Note 1

vector creative for $\vec(a)$ i $\vec(b)$ є $\vec(c)$, which is the third vector $\vec(c)= ||$, moreover, this vector can be especially powerful:

The scalar of the subtracted vector is the extension of $|\vec(a)|$ i $|\vec(b)|$ by the sine of the cut $\vec(c)= ||= |\vec(a)| \cdot |\vec(b)|\cdot \sin α \left(1\right)$;
All $\vec(a), \vec(b)$ and $\vec(c)$ satisfy the triplet;
The subtraction vector is orthogonal to $\vec(a)$ i $\vec(b)$.

As for the vector in the presence of the coordinates ($\vec(a)=$x_1; y_1; z_1$$ i $\vec(b)= $x_2; y_2; z_2$$), then coordinate systems can be determined by the formula:

$ = $y_1 \cdot z_2 - y_2 \cdot z_1; z_1 \cdot x_2 - z_2 \cdot x_1; x_2 \cdot y_2 - x_2 \cdot y_1$$

It is easier to remember the formula by writing it down in the form of the signer:

$ = \begin(array) (|ccc|) i&j&k\x_1&y_1&z_1\\x_2&y_2&z_2\\end(array)$.

Tsya formula is already familiar for vikoristannya, but in order to understand, how to vikoristovuvat, on the back of the head, you should become familiar with the theme of the matrix and їх vyznachnіv.

The area of the parallelogram, the sides of which are defined by two vectors $\vec(a)$ and $vec(b)$ scalar of vector creation of given two vectors.

Tse spіvvіdnoshennia duzhe easily vesti.

Let's guess the formula for knowing the area of a splendid parallelogram, which can be characterized by the brackets $a$ and $b$:

$S = a \cdot b \cdot \sin α$

For whichever side the scalar values of the vectors $\vec(a)$ and $\vec(b)$ are more suitable for us, then the scalar of the vector creation of these vectors will be the plane of the figure.

butt 1

Given a vector $\vec(c)$ with coordinates $$5;3; 7$$ and a vector $\vec(g)$ with coordinates $$3; 7;10 $$ in the Cartesian coordinate system. Find out why the area of the parallelogram made $\vec(c)$ and $\vec(g)$ is worthy.

Solution:

We know the vector TV for these vectors:

$ = \begin(array) (|ccc|) i & j & k \\ 5 & 3 & 7 \\ 3 & 7 & 10 \\ \end(array)= i \cdot \begin(array) (|cc |) 3 & 7 7 & 10 \end(array) - j \cdot \begin(array) (|cc|) 5 & 7 \\ 3 & 10 \end(array) + k \cdot \begin(array) ( |cc|) 5 & 3 \\ 3 & 7 \\ \end(array) = i \cdot (3 \cdot 10 - 49) - j \cdot (50 -21) + k \cdot (35-9) = -19i -29j + 26k = $-19;29;26$$.

Now we know the modulo value for the removed straightened wedge, but the values of the area of the induced parallelogram:

$S= \sqrt(|19|^2 + |29|^2 + |26|^2) = \sqrt(1878) ≈ 43.34$.

This crossing of the world is just not only for the recognition of the area in the 3-world expanse, but also for the 2-world space. Get to know the upcoming tasks on this topic.

butt 2

Calculate the area of the parallelogram, so as to make it possible, the edges are given by the vectors $\vec(m)$ with the coordinates $$2; 3$$ and $\vec(d)$ with the coordinates $$-5; 6$$.

Solution:

This task is a private example of problem 1, it's better, but if it's offending, the vectors lie in the same plane, and tse means that the third coordinate, $ z $, can be taken as zero.

According to the foregoing, the square of the parallelogram stock:

$S = \begin(array) (||cc||) 2 & 3\ -5 & 6 \\ \end(array) = \sqrt(12 + 15) =3 \sqrt3$.

butt 3

Given a vector $\vec(a) = 3i - j + k; \Vec(b)=5i$. Appreciate the area of the parallelogram established by them.

$[ \vec(a) \times \vec(b)] = (3i - j + k) \times 5i = 15 - 5 + $

It’s easy to see with the induced table for single vectors:

Figure 1. Decomposition of a vector behind a basis. Author24 - Internet exchange of student works

$[ \vec(a) \times \vec(b)] = 5 k + 5 j$.

Hour of pidrakhunkiv:

$S = \sqrt(|-5|^2 + |5|^2) = 5\sqrt(2)$.

The previous tasks were about vectors, the coordinates of some tasks in the Cartesian coordinate system, but we can also look at the difference between the basis vectors in $90°$:

butt 4

The vector $\vec(d) = 2a + 3b$, $\vec(f)= a – 4b$, if $\vec(a)$ and $\vec(b)$ are equal to each other and equal to one, and between $\vec(a)$ and $\vec(b)$ 45°.

Solution:

We can calculate the vector TV $\vec(d) \times \vec(f)$:

$[\vec(d) \times \vec(f) ]= (2a + 3b) \times (a - 4b) = 2 - 8 + 3 - 12 $.

For vector creations, it is right to attack with their powers: $$ and $$ equal to zero, $ = - $.

Vikoristovuemo tse for forgiveness:

$[\vec(d) \times \vec(f)] = -8 + 3 = -8 - 3 = -11 $.

Now we accelerate with the formula $(1)$ :

$[\vec(d) \times \vec(f)] = |-11 | = 11 \cdot |a| \cdot |b| \cdot \sin α = 11 \cdot 1 \cdot 1 \cdot \frac12=5.5$.

The area of the parallelogram, based on vectors, allows dobutku dozhin tsikh vectorіv on kut kuta, which lie between them.

Good, if for the minds of the minds of these vectors. However, it is so, that the formula for the area of a parallelogram, based on vectors, can only be calculated after the coordinates have been drawn.
It’s a blessing, and for the minds it’s given a lot of vectors, it’s just necessary to fill in the formula, which we have already reportedly sorted out in the stats. The area for adding additional modules for the sinus cut between them:

Let's take a look at the butt of the rozrahunka of the area of the parallelogram, based on vectors.

Manager: the parallelogram of impulses on the vectors ta . Know the area, yakscho, and cut between them 30 °.
Virasimo vector through їх values:

Possibly, you have a winiclo diet - the stars came from zero? Varto guess what works for vectors, and for them . so to bring respect, that as a result we will take viraz, then it will be converted to. Now we are carrying out the sum of the sums of money:

Let's turn to the problem, if the vector vector is not shown in the minds. If your parallelogram lies near the Cartesian coordinate system, then it is necessary to do so.

Rozrahunok of the dozhin of the sides of the figure, given by the coordinates

For the cob, we know the coordinates of the vectors and we can see the coordinates of the cob and the coordinates of the cob. It is possible to coordinate the vector a (x1; y1; z1), and the vector b (x3; y3; z3).
Now we know the length of the skin vector. For this skin coordinate, it is necessary to square it, then add up the subtracted results and take the root from the final number. Behind our vectors there will be coming roses:

Now it is necessary to know the scalar reality of our vectors. For each of them, the coordinates are multiplied and added.

We can know the cosine of the coota that lies between them .
Now we can know the sine of which well kut:
Now we have all the necessary quantities, and we can easily know the area of the parallelogram, based on the vectors for the already known formula.

On this level, we can look at two more operations with vectors: vector booth vector_vі Zmіshany tvіr vectorіv (Vіdrazu possilannya, who needs the very thing). It's nothing terrible, so sometimes it's just for total happiness, krim scalar creative vector , Need more and more. This is the vector axis of drug addiction. Might add up an animosity that we can climb into the net of analytic geometry. Tse not so. For whom the great mathematicians have taken little firewood, it’s better to hang out on Pinocchio. Really, the material is more wide and simple - hardly more foldable, lower than the same scalar doboot , there will be less typical tasks. Golovne in analytic geometry, like a lot of people who change their minds and already having a mess, DO NOT HAVE MERCY IN HIVISLE. Repeat like a spell, and you will be happy.

Like vectors and vibrate here far away, like glitters on the horizon, don’t be, start from the lesson Vectors for teapots , in order to learn or to gain basic knowledge about vectors. Readers can learn more about this information, I have tried to collect as much as possible a collection of applications, which are often used by practical robots

What will make you happy? If I'm small, then I've learned to juggling two and wrapping three in bags. It was creepy. At the same time, juggling will not happen in a flash, the shards of our eyes can be seen only space vectors, and the flat vectors from two coordinates are left behind. Why? This is how the data were already born - the vector is not the same zmіshane tvіr vektorіv is designated to practice in the trivial space. Already easier!

In this operation, just like in a scalar creation, take part two vectors. Let there be immortal letters.

diya herself be appointed let's come in rank: . Іsnuyut and іnshі options, but I also use the sound to designate a vector tvir vector in the same way, in square arms with a cross.

I immediately food: yakscho in scalar creation of vectors take the fate of two vectors, and here also multiply two vectors, then what difference? Clear difference, first for everything, as a RESULT:

The result of the scalar vector creation is є:

VECTOR: , then the vector is multiplied and the vector is taken again. Closed club. Vlasne, the sound is the name of the operation. In different primary literature, the meaning of the same can be changed, I choose the letter .

Designation of vector creation

I'll be back with a picture, then comments.

Appointment: Vector creative noncollinear vectoriv, taken from given order, called VECTOR, dozhina numerically better area of the parallelogram, based on these vectors; vector orthogonal to vectors, and directings so that the basis has the right orientation:

We choose the appointment by the brushes, there is a lot of ringing here!

Again, you can name the following moments:

1) Outside vectors, marked with red arrows, for the designated not collinear. Vipadok kolіnearnyh vektor_v before the river will look trohi pіznіshe.

2) Take vectors in a strictly defined order: – "a" multiplied by "be", and chi is not "be" to "a". The result of the multiplication of vectorsє A vector with blue color values. If you multiply the vectors y in reverse order, then we take away the vector equal to the distance and the straight vector (crimson color). Tobto fair jealousy .

3) Now cognizable from the geometric zm_st vector creation. This is an extremely important point! The length of the blue vector (and, also, i of the crimson vector) is numerically greater than the area of the parallelogram, based on the vectors. On the little one is a parallelogram of shading with black color.

Note : armchair є schematic, і, naturally, the nominal value of the vector creation is not equal to the area of the parallelogram.

We guess one of the geometric formulas: the area of the parallelogram is more expensive to add the sum of the sides to the sine of the cut between them. To that, according to the foregoing, the formula for calculating the DOVZHINI of the Vector creation is valid:

I reiterate that the formulas have about the DOWN of the vector, and not about the vector itself. What practical zmist? And the sense is such that the definition of the analytical geometry of the area of a parallelogram is often known through the concept of a vector product:

Let's take a friend an important formula. The diagonal of the parallelogram (black dotted line) divides the yogo into two equal tricots. Later, the area of the tricutnik, inspired by vectors (black shading), can be known by the formula:

4) No less important fact is that the vector is orthogonal to the vectors, that . Understandably, the straightening vector (crimson arrow) is also orthogonal to the outward vectors.

5) The vector of straightening so that basis may law orientation. On the lesson about go to a new basis I report back about plane orientation and at once we will figure out what kind of orientation to space. I will explain on your fingers right hand. Think about it eye-catching finger with vector i middle finger with a vector. Ring finger and little finger press down to the valley. As a result thumb- Vector tvir is uphill. Price and є right-orientation basis (on a small scale of faults). Now remember the vectors ( expressive and middle fingers) by the hands, as a result, the thumb will flare up, and the vector tvir will already move down. This is also a right-orientation basis. Possibly, you have a winklo of food: what kind of basis can I have a left orientation? "Invite" the same fingers left hand vectors , and take away the left basis and the left orientation of the space (in my case, the great finger is spread out at the straight line of the lower vector). Figuratively, apparently, the bases “twist” or orient the space at the different sides. And it’s not easy to understand if we think up something abstract - so, for example, the orientation of the space changes the size of the mirror, and it’s like “strike the object from the mirror”, then you can’t go into the wild with "original". Before speech, put three fingers to the mirror and analyze the impression;-)

... it’s still good, what do you now know about right and left orientation bases, more scary talk of such lecturers about changing orientation =)

Vector tvir of collinear vectors

Appointment is reported to the branch, there is no more explanation, what is needed, if the vectors are collinear. As vectors are collinear, then they can be expanded on one straight line and our parallelogram can also be folded into one straight line. Such an area, as it seems to be mathematicians, virogenous The parallelogram is equal to zero. Tse w vyplivaє i z formulas - the sine of zero or 180 degrees to zero, and therefore, the square of zero

In such a rank, yakscho, then і . To pay attention to the fact that the vector itself is equal to the zero vector, but in practice it is often difficult to write that the vector is also equal to zero.

Okremy vipadok - vector tvir of the vector on itself:

For the help of the vector creation, the collinearness of the trivi- mer vectors can be reversed, and the task of the middle of the other conflicts can be sorted out.

For the perfection of practical applications, you may need trigonometric table , to find the meaning of sinuses.

Well, let's fire the fire:

butt 1

a) Know the value of the vector creation of vectors, so

b) Find the area of the parallelogram based on the vectors

Solution: Hі, tse not a drukarska pardon, vihіdnі danі in the points of the mind, I navmisno zrobiv the same. That's why the design decision is taken care of!

a) It is necessary for the mind to know dozhina vector (vector creation). For a specific formula:

Vidpovid:

If you ate about dovzhina, then it seems that you are showing peace - loneliness.

b) It is necessary for the mind to know area a parallelogram based on vectors. The area of this parallelogram is numerically superior to the vector creation:

Vidpovid:

To give respect to the fact that there are no warnings about vector witting, we were inquired about square figures vіdpovіdno rozіrnіst - kvadnі odinіtsі.

Always marvel at what it is necessary to know beyond the mind clear proof. You can do it with letters, ale letters in the middle of vikladachiv vistacha, and with good chances to turn around for additional treatment. Although the reasoning is not particularly strained - if it is not correct, then there is a reaction that the person does not understand in simple speeches and / or does not delve into the essence of the task. At this moment, you need to try on the control, virishuyuchi be-like zavdannya z mathematician and z іnshih subjects tezh.

Where did the great letter "en" go? In principle, її it was possible to stick to the decision, but with the method of speeding up the recording, I didn’t kill it. I spodіvayus, all zrozumіlo, scho and tse signification of one and the same.

A popular butt for independent vision:

butt 2

Know the area of trikutnik, inspired by vectors, yakscho

The formula for the area of the tricot through the vector dobutok is given in the comments before the appointment. The solution is to follow the example of the lesson.

In fact, the dressing room is really wide, they can roll it up with tricots.

For the accomplishment of other tasks, we need:

The power of the vector creative vector

We already looked at the leaders of the authority of the vector creation, I will include them in the list.

For more vectors and a greater number, the following powers are valid:

1) In other sources of information, this item is not heard by authorities, but it is still important in practical terms. So let it be.

2) - Power tezh rozіbrano more, іnоdі yogo call anticommutative. Otherwise, apparently, the order of the vector may be significant.

3) - happy or associative laws of vector practice. Konstanty seamlessly blame for intervector creativity. Really, what do they need to do?

4) - rozpodіlnі abo distributive laws of vector practice. There are also no problems for opening the shackle.

As a demonstration, a short butt is looked at:

butt 3

Know yakscho

Solution: For the mind, it is necessary to know the realm of the vector creation. Let's write our miniature:

(1) Zgіdno z associative laws, we blame the constant for intervector creation.

(2) We blame the inter-module constant, its own module has a “minus” sign. Dovzhina can be negative.

(3) I understood further.

Vidpovid:

The hour has come to add firewood to the fire:

butt 4

Calculate the area of the trickster, inspired by vectors, as

Solution: The area of \u200b\u200bthe trikutnik is known by the formula . The catch is that the vectors "ce" and "de" themselves are represented as a sum of vectors. The algorithm here is standard and guess what, apply No. 3 and 4 to lesson Scalar tvir vector_v . For clarity, the solution is divided into three stages:

1) On the first crochet, we can see the vector tvir through the vector tvir, in fact, virazimo vector through vector. About dozhini still no words!

(1) Represented by a number of vectors.

(2) Vikoristovuyuchi distributive laws, opening the arches for the rule of multiplication of rich terms.

(3) Vikoristovuyuchi associative law, we blame all the constants for intervector creations. With a small dosvіdі dії 2 і 3 it is possible to beat one hour.

(4) First and foremost, the rest of the additions to zero (zero vector) are the rewards of receiving power. Another addendum has the power of anticommutativity of the vector creation:

(5) Suggest similar dodanki.

As a result, the vector appeared through the vector, which is necessary to achieve:

2) At another stage, we will know the length of the vector creation we need. Tsya deya guessing Butt 3:

3) We know the area of the shukan tricoutnik:

Stages 2-3 solutions can be completed in one row.

Vidpovid:

Take a look at the task to make it wider in the control robots, the axis of the butt for an independent variance:

butt 5

Know yakscho

Briefly, the solution is to illustrate the lesson. Surprisingly, how much you were respectful of the front butts ;-)

Vector tvіr vectorіv y coordinates

, given in the orthonormal basis , expressed by the formula:

The formula is really simple: at the top row of the signifier, coordinate vectors are written, at the other and third rows, the coordinates of the vectors are “stacked”, moreover, it is in strict order- First coordinates of the "ve" vector, then coordinates of the "double-ve" vector. If vectors need to be multiplied in a different order, then the rows should be remembered as spaces:

butt 10

Verify, what are the next vectors and space:
A)
b)

Solution: The revision is based on one of the principles of this lesson: since vectors are collinear, then their vector complement is equal to zero (zero vector): .

a) We know the vector TV:

In this manner, the vectors are not collinear.

b) We know the vector TV:

Vidpovid: a) not collinear; b)

Axis, maybe, and all the main information about vector creation of vectors.

Tsej rasdіl bude small, oskolki zavdan, de vikoristovuetsya zmіshane tvіr vektorіv, not rich. Practically everything will fit into the design, geometrical change and sprat of working formulas.

Zmishany TV vector:

The axis stinks so much like a train and check, do not check, if they are charged.

On the back of my head, I’ll rediscover that picture:

Appointment: Created with creativity non-coplanar vectoriv, taken from given order, called obsyag paralepiped, based on these vectors, with the “+” sign, so the basis is right, and the “–” sign, so the basis is left.

We see the little ones. The lines invisible to us are crossed with a dotted line:

Zanuryuёmosya at the appointment:

2) Take vectors in song order, so the permutation of vectors in creation, as you guess, does not pass without traces.

3) Before that, as a commentary on a geometrical change, I will state an obvious fact: zm_shany tv_r vectorіv є NUMBER: . In the initial literature, the design can be somehow different, I mean the sound is zmishane tvir through, and the result is calculated with the letter “ne”.

For appointment zmіshany tvіr - tse obsyag paralelepiped, based on vectors (the figure is crossed with red vectors and black color lines). That is the number of the old obyagu of this parallelepiped.

Note : chairs are sketchy.

4) Don't try again to understand the orientation of the basis and space. The sense of the final part of the one who can take the obligatory sign is minus. In simple words, zmishane tvir can be negative: .

The following is a formula for calculating the volume of a parallelepiped based on vectors.

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