Area parallelogram prompted by vectors numbered yak dobutok dozhin qih vectors on the sinus kuta between them. If we do not see the coordinates of the vectors, for the calculation it is necessary to set the coordinate methods, the growth and the designation of the cut with vectors.

You know

• - Understanding the vector;
• - the power of vectors;
• - Cartesian coordinates;
• - Trigonometric functions.

Instructions

• If you see more vectors and cuts between them, then in order to know the area parallelogram prompted by vectors, know a number of їх modules (additional vectors), on the sine of a cut between them S = │a│ │ b│ sin (α).
• If the vectors are set in the Cartesian coordinate system, then in order to know the area parallelogram prompted by them, visit the following:
• Know the coordinates of the vectors, if the stench is not given at once, when you see the coordinates of the vectors, the coordinates of the cob. For example, if the coordinates of the cob point of the vector are (1; -3; 2), and the end point (2; -4; -5), then the coordinates of the vector will be (2-1; -4 + 3; -5-2) = (1 ; -1; -7). Find the coordinates of vector a (x1; y1; z1), vector b (x2; y2; z2).
• Know the amount of skin vector. Make the skin from the coordinates of the vectors in the square, know the amount x1² + y1² + z1². The result is a square root. For another vector, use the same procedure. With such a rank, viide │a│in b│.
• Know the scalar addition of vectors. For the whole, multiply the given coordinates and put the creation │a b│ = x1 x2 + y1 y2 + z1 z2.
• Start the cosine cut between them for which scalar add-on vectors, for which the scalar add-on vectors are added, for example, at item 3, add to the add-on vectors, as the bullets are rooted in item 2 (Cos (α) = │a b│ / (│a│ │ b│)).
• The sine of the cut-off kut will be the square root at the difference of the number 1, and the square of the cosine of the same cut, rooted in p. 4 (1-Cos² (α)).
• Crack the area parallelogram prompted by vectors Knowing dobutok їkh dovzhin, calculating in item 2, and multiply the result by the number that went out of the list at item 5.
• While the coordinates of the vectors are given on the area, when the razrakhunks, the z coordinate is simply displayed. Tsey rozrahunok є numerical viraz of a vector add two vectors.

The area of ​​the parallelogram, impelled by vectors, brings you a couple of dozens of vectors on the cube of a kut, so that you can lie between them.

Good, if the minds are given more vectors. However, you can only write the formula for the area of ​​the parallelogram induced on vectors by writing the coordinates for the coordinates.
I was spared, and for the minds were given more vectors, you just need to find a formula, as they were already sorting out in detail in the statistics. The area for the additional modules on the sinus cut between them:

The butt of the plots of the parallelogram, prompted on vectors, is visible.

Zavdannya: parallelogram of incentives on vectors ta. Know the area, where, and the cut between them is 30 °.
Virazimo vector through їх value:

Mozhlivo, have you got food - the stars have come from zero? Varto zgadati, with mi pratsyєmo with vectors, and for them ... so it’s a beast of respect, that if in the results we’ll be able to recognize it, then it’s going to be transformed into. Now the following are carried out:

Turning to the problem, if more vectors are not indicated for the minds. As long as your parallelogram lies in the Cartesian coordinate systems, then it will be necessary to change it as well.

Rozrakhunok dovzhin siden figuri, given by coordinates

For the cob, the coordinates of the vectors and from the coordinates of the final coordinates of the cob are known. Acceptable coordinates of vector a (x1; y1; z1), and vector b (x3; y3; z3).
Now we know more than the genius of the skin vector. For the whole skin coordinate, it is necessary to give a square, after the edges of the cutoff the results and the end of the number of roots. Behind our vectors there will be advances:


Now it is necessary to know the scalar tvir of our vectors. For many of the given coordinates, multiply and store.

Mayuchi dozhini vectors and їх scalar tvir, we can know the cosine of a cut, how to lie between them .
Now we can know the sine of this kuta:
Now we have all the necessary quantities, and we can easily know the area of ​​the parallelogram prompted on the vectors by the same formula.

At all levels, there are two operations with vectors: vector dobutok vectorsі change tvir vectors (as soon as possible, who needs it)... Nichogo terrible, so inodi boo, well for general happiness, krim scalar vectors, Needed more and more. This is the axis of vector drug addiction. The feud may be curled up in the absence of analytical geometries. It's not like that. The razdіlі of vishoї mathematics has not enough firewood, it is necessary to work on Buratino. As a matter of fact, the material is even worse than extensions and simpler - it is unlikely that it is more foldable, not the same scalar tvir, there will be less typical buildings. The head is in analytical geometry, as it is a lot of things to get over and over again, you will not be merciful to HIV-positive people. Repeat the yak spell, if you have happiness =)

Yakshto vektori be here far away Vektori for teapots, you should know about vectors. More cooked readers can learn about the information vibrating, I will try to increase the number of butts as much as possible, as I often use practical robots

How can I make you happy? If I am small, then I can juggle two and wind three bags. It was spontaneous. Infection jugglyuvati will not happen in the zagala only open spaces vector, and the plane vectors from two coordinates go overboard. For what? Such are already the birth of the diii - the vector and the change of the tv vector in the designation of the trivial space. As simple as that!

In the whole operation, so it goes, like in the scalar creation, take on the fate two vectors... Let there be no letters.

The very dia signify step by step:. Find the best options, or even the sound of the vector twir of the vectors in the same way, at the square arches from the cross.

First and foremost nourishment: yaksho in scalar creation of vectors take the fate of two vectors, і there can be multiplied by two vectors, todі why growth? Yavna growth, persh for everything, IN RESULTS:

The result of the scalar vector vectors є NUMBER:

The result of the vector vector VECTOR:, so that the vector is multiplied and the vector is known. Close the club. Vlasne, the name of the opera is. In the development of the new literature, the meaning can be changed, I will be victorious with the letter.

The value of the vector creation

There will be a small selection of pictures, then some comments.

Viznachennya: Vector cheese non-collinear vector_v, taking from this order, be called VECTOR, dovzhina how numerically road parallelogram areas motivated by given vectors; vector orthogonal vectors, and conjugations so that the basis can be rightly arranged:

Picked up by hand, there is a lot of color here!

Also, you can see these sutta moments:

1) Vyhіdni vectors, denoted by red arrows, by viznenny not collinear... The variety of collinear vectors will be clearly visible.

2) The vectors are taken in a strictly assigned order: – "a" multiplied by "ba", and chi is not "be" to "a". The result of multiple vectorsє VECTOR, which means blue color. If the vector is multiplied in the vortex order, we can distinguish the rivny for the female and the opposite for the straight vector (raspberry color). Tobto fair parity .

3) Now cognizable from the geometric snake of the vector creation. This is a very important point! DOSE of the blue vector (a, also, і of the raspberry vector) is numerically the size of the AREA of the parallelogram induced on the vectors. On a tiny little parallelogram of shading with black color.

Note : armchair є schematic, і, naturally, it is nominal for the vector art not for the area of ​​the parallelogram.

Guess one of the geometric formulas: the area of ​​the parallelogram of the road to the addition of the summed sides on the sinus of the kuta between them... To the one who is in love with what has been said, the formula for calculating DOVE vector is valid:

I wonder if the formula is about the TRUE of the vector, and not about the vector itself. What a practical wolf? And the sense is such, that the staff of analytical geometries often know the area of ​​the parallelogram through the understanding of the vector creation:

An important formula for a friend. The diagonal of the parallelogram (red dotted line) should be divided into two equal tricytes. Otzhe, the area of ​​the tricycle, prompted by vectors (red shading), can be found behind the formula:

4) The fact that the field is not less important is that the vector is orthogonal to the vectors, so that ... Zrozumіlo, oppositely rectifying vector (raspberry arrow) is also orthogonal to the output vectors.

5) Vector of conjugations so, scho basis maє right orієntаtsіyu. At the lesson about transitions to a new basis I have finished reporting reports about orієntatsії area and at once they are free, so also the freedom to open space. I will explain on your fingers right hands... Find some thoughts last finger with vector i middle finger with a vector. Ring finger and little squeeze to the valley. As a result thumb- Vector TV will be amazed up the hill. Tse і є legal basis basis (for a little bit itself). Now remember the vectors ( middle finger) In a few seconds, as a result, the great finger will flare up, and the vector TV will already be astonished downward. Tse is also a legal basis. Mozhlivo, you have a dietary fault: what basis is there for less understanding? "Attract" to the same fingers left hands vector, і trim the lіviy basis і lіvu аrієntatsіyu space (in tsyomu vipadku the great toe spread out at the right edge of the lower vector)... Figuratively, it seems that the bases "twist" or the space around the sides. The first understanding does not mean to respect what we think of as abstract - so, for example, to the open space of the zvichaynisinke mirror, as well as the "vitiagti see the object from the mirror" Before the speech, go to the mirror three fingers and analyze the image ;-)

... yak, after all, it’s good, now you know about right- and lіvoorієntovanih bases, more terrible vislovuvannya deyak lecturers about the change of thinking =)

Vector twir of collinear vectors

The date of the report has been selected, it has become too much of a problem, it’s possible to see it, if there are collinear vectors. As the vector is collinear, it is possible to expand it on one straight line, and our parallelogram can be "folded" into one straight line. The area of ​​such, as it seems to be mathematics, virogen Parallelogram to zero. This is the sine of zero or 180 degrees to zero, which means that the area is zero

Such a rank, yaksho, then ... Strictly it seems, the vector addition itself to the zero vector, but in practice it is often not necessary to write and write, but it is just that it should be zero.

Particle vipadok - vector addition of a vector on itself:

For the additional vector creation, you can reverse the number of trivial vectors, and the process of setting the middle ones can be freely selected.

For practical applications you can use it trigonometric table, the schob knows the values ​​of the sinuses.

Well, well, rozpalyєmo fire:

Butt 1

a) Know the genius of the vector vector

b) Know the area of ​​the parallelogram induced on vectors, if

Decision: Hi, it’s not a drukarska pomilka, vikhіdnі danі in the points of mind, I navmino shattered the same. That’s why the design decision will come out!

a) It is necessary to know for the mind to dinner vector (vector to yours). For a general formula:

View:

If I was fed up with dinner, then it seems that the size is one.

b) It is necessary to know for the mind area a paralelogram induced on vectors. The area of ​​the given parallelogram is numerically significant for the vector addition:

View:

Beastly respect, well, the news about the vector TV does not go astray, we were fed about figuri areas according to the size - square units.

Become astonished, it is necessary to know what is behind the mind, and go out from the formula clear view. You can start with literalism, ale of letters in the middle of viclades of vistachaє, and with good chances to turn to additional optimization. If the trick is not particularly strained - if it seems incorrect, then there will be hostility, but the people do not mind in simple speeches and / for not grasping the essence of the envy. The whole moment you need to trim on the control, be sure to learn from the whole of mathematics and from other subjects, like.

Where did the big letter "en" go? As a matter of principle, it is possible to adhere to the point before the decision, even with a quick note of speed, I’m not broken. I am encouraged, with all the intelligence, that is the meaning of one and the same.

Popular buttstock for self-determination:

Butt 2

Know the area of ​​a tricycle driven by vectors, yaksho

The formula for determining the area of ​​a tricycle in terms of vector addition is given in the comments to the date. Decision and suggestion for a lesson.

At the practical level, the fair is wider, the tricytes can go down.

For the latest news, we know:

The power of vector creation vectors

The deyak of the power of the vector creation has already been looked at, I will include the whole list.

For a fair number of vectors and a fair number, the following powers are true:

1) In their information dzherels, the point is not seen by the authorities, but even more important to the practical plan. Also, don't bother.

2) - the power of the same name anticomutative... As it seems, the order of the vectors is significant.

3) - single abo associative laws of vector pratsi. Constant has no problem to blame for the boundaries of the vector creation. Really, who is it?

4) - rozpodilny abo distribution laws of vector pratsi. There are also no problems with the opening of the temples.

I will demonstrate a short butt for a demonstration:

Butt 3

Know yaksho

Decision: For clever knowledge, it is necessary to know the amount of vector art. Describe our miniature:

(1) According to the associative laws, the blame is constant beyond the vector creation.

(2) We are guilty of the constant between the modulus, the "minus" sign has its own module "z'ydag". Dovzhina can be negative.

(3) Farther farther.

View:

It's time to drop firewood by the fire:

Butt 4

Calculate the area of ​​a tricycle driven by vectors, yaksho

Decision: The area of ​​the tricycle is known for the formula ... The catch is that the vector "tse" and "de" itself are represented as a sum of vectors. The algorithm here is standard і chimos nagaduє butt No. 3 and 4 lesson Scalar add-on vectors... The solution for clarity is rosib'єmo in three stages:

1) On the first small scale, the vector tvir through the vector tvir, by day, virasimo vector through vector... Leave no words about dozhini!

(1) Introduce a vector virazi.

(2) Vikoristovuchi distributive laws, open arms for the rule of multiple bugs.

(3) Vikoristovuchi associative laws, blame all the constants for the inter-vector creations. With a small amount of information from 2 to 3, you can visit one hour.

(4) The first and last addition to the zero (zero vector) of the establishment of the acceptance of power. The other side of the Vikorist has the power of anticomutativity of the vector creation:

(5) Probably a little extra.

As a result, the vector appears through the vector, which must be reached:

2) At the other stage, we know the amount of vector creation we need. Tsya diya nagaduє Appendix 3:

3) We know the area of ​​the shukany tricycle:

Step 2-3 solutions can be issued in one row.

View:

Take a look at the width of the control robots, the butt axis for the independent version:

Butt 5

Know yaksho

A short solution and a summary of the lesson. Surprisingly, we have a lot of respectable butts in front of them ;-)

Vector twir of vectors in coordinates

given in an orthonormal basis, swing the formula:

The formula is simple: at the top row of the formatting tool, the coordinate vectors are written, at the other and third rows, the coordinates of the vectors are written, and the contribution is have a strict order- I'll take the coordinates of the vector "ve", then the coordinates of the vector "double-ve". If the vectors need to be multiplied in the same order, then the rows should be remembered in mice:

Butt 10

Revise, where collinear will be on the way to the vastness:
a)
b)

Decision: The revision is based on one of the instructions given to the lesson: if the vector is collinear, then the vector add-on goes to zero (to the zero vector): .

a) We know the vector tvir:

In such a rank, the vectors are not collinear.

b) We know the vector tvir:

View: a) not collinear, b)

Axis, mabut, and all basic views about vector add-on vectors.

Daniy will be no worse than great, oskіlki zavdan, de vikoristovutsya zmіshane tvіr vectors, nebagato. In fact, everything fits into a value, a geometric change and a couple of working formulas.

Zm_shaniy tvir vectors - tse tvir three vectors:

The axle stinks like a locomotive and checks, it’s not dying, if it’s counted.

Check out the following picture:

Viznachennya: Zmіshanim cheese non-coplanar vector_v, taking from this order, be called ob'єm parallelepipeda, prompted on given vectors, with the sign "+", which is the basis of the rules, and the sign "-", which is the basis of the lines.

Viconaєmo babies. The lines are invisible to us with a dotted line:

Porinaєmo at viznachennya:

2) The vectors are taken singing order, so that the rearrangement of the vectors in the creature, as you will, you will not be mined without the inheritance.

3) Before that, as a commentary on the geometric snake, I mean an obvious fact: change of vectors є NUMBER:. At the beginning of the literature, the design can be made very often, I know the sound of the change through, and the result is numbered in the letter "pe".

For viznachennyam change tvir - tse obsyag parallelepipeda, prompted on vectors (the figure is covered with red vectors and lines of the black color). That is the number of the last time a given parallelepiped.

Note : armchair є schematic.

4) We will not know how to soar with the understanding of the basis and space. The sense of the final part of the one who can give a minus sign to the debate. In simple words, the change of tvir can be negative:.

Bezposeredno s value next is the formula for calculating the amount of parallelepiped, prompted on vectors.