At this lesson, we will consider two more operations with vectors: vector artwork vectors and mixed vectors (immediately link, who needs it. Nothing terrible, so sometimes it happens that for complete happiness, in addition scalar product vectors, it is also required. Such here is a vector drug addiction. It may seek the impression that we climb into the debris of analytical geometry. This is not true. In this section of the highest mathematics, there is not enough firewood in general, except for Pinocchio. In fact, the material is very common and simple - hardly more difficult than the same scalar productEven the typical tasks will be smaller. The main thing in analytical geometry, as many will be killed or have already been convinced, not mistaken in calculations. Repeat as a spell, and you will be happy \u003d)

If the vectors sparkle somewhere far as lightning on the horizon, not trouble, start from the lesson Vectors for teapotsTo restore or re-acquire basic knowledge about the vectors. More prepared readers can get acquainted with the information selectively, I tried to collect the most complete collection of examples that are often found in practical work.

What you immediately please? When I was small, then I knew how to juggle two and even three balls. Deftly succeeded. Now you will not have to juggle at all, as we will consider only spatial vectors, and flat vectors with two coordinates will remain overboard. Why? These actions were born - a vector and mixed product of vectors are defined and operated in three-dimensional space. Already easier!

In this operation, in the same way as in the scalar product, participate two vectors. Let it be nonsense letters.

Action itself denotes in the following way: . There are other options, but I used to denote the vector artwork of vectors just like that in square brackets with a cross.

And immediately question: if in scalar product vectors Two vectors are involved, and here two versions are multiplied here, then what is the difference? Explicit difference, first of all, as a result:

The result of a scalar product of vectors is a number:

The result of vector art vectors is vector:, That is, we multiply the vectors and get the vector again. Closed club. Actually, hence the name of the operation. In various learning literature, the designations can also vary, I will use the letter.

Definition of vector art

First there will be a definition with a picture, then comments.

Definition: Vector work nonollyline vectors, taken in this order, called vector, length which is numerical equal to the square of the parallelogrambuilt on these vector data; vector orthogonal vectors And it is directed so that the basis has the right orientation:

We disassemble the definition of the bones, there is a lot of interesting things!

So, you can select the following essential moments:

1) source vectors marked with red arrows by definition not collinear. The case of collinear vectors will be appropriate to consider a little later.

2) vectors taken in strictly defined order: – "A" is multiplied by "BE", not "BE" on "A". The result of multiplication vectors It is a vector that is indicated in blue. If the vectors are multiplied in reverse order, then we get equal to the length and the opposite vector (raspberry color). That is, equality is right .

3) Now let's get acquainted with the geometric meaning of the vector product. This is a very important point! The length of the blue vector (and, therefore, the raspberry vector) is numerically equal to the square of the parallelogram built in the vectors. In the figure, this parallelogram is shaded in black.

Note : The drawing is schematic, and naturally, the nominal length of the vector product is not equal to the area of \u200b\u200bthe parallelogram.

We remember one of the geometric formulas: the area of \u200b\u200bthe parallelogram is equal to the product of adjacent sides on the corner sine between them. Therefore, based on the foregoing, the formula for calculating the length of the vector product:

I emphasize that in the formula we are talking about the length of the vector, and not about the very vector. What is the practical meaning? And the meaning is that in the tasks of the analytical geometry, the area of \u200b\u200bthe parallelogram is often found through the concept of vector art:

We will get a second important formula. The diagonal of the parallelogram (red dottedier) divides it into two equal triangles. Consequently, the area of \u200b\u200bthe triangle, built in the vectors (red hatching), can be found by the formula:

4) no less important fact is that the vector is orthogonal vectors, that is . Of course, the oppositely directed vector (the raspberry arrow) is also orthogonal in the original vectors.

5) the vector is directed so that basis It has right Orientation. In the classroom O. transition to a new basis I spoke in detail about orientation of the planeAnd now we will deal with the orientation of space. I will explain to your fingers right hand. Mentally combine forefinger With vector I. middle finger With vector. Unnamed finger and a little finger Press the palm. As a result thumb - vector art will look up. This is the right-fledged basis (in the figure it is he). Now change the vectors ( index and middle fingers) Places, as a result, the thumb unfolds, and the vector work will already look down. This is also a regular basis basis. Perhaps you have a question: what basis does the left orientation? "Name" the same fingers left hand vectors and get the left basis and the left orientation of the space (In this case, the thumb will be located in the direction of the lower vector). Figuratively speaking, these bases "spin" or orient space in different directions. And this concept should not be considered something contrived or abstract - so, for example, the orientation of the space changes the most ordinary mirror, and if you "pull the reflected object from the castorcal." It will not be able to combine it in general. By the way, bring three fingers to the mirror and analyze the reflection ;-)

... how it's good that you now know about law and left-oriented Bases, for the terrible statements of some lecturers about the change of orientation \u003d)

Vector artwork of collinear vectors

The definition disassembled in detail, it remains to find out what is happening when the collinear vectors. If the vectors are collinear, then they can be placed on one straight line and our parallelogram also "folds" into one straight. The area of \u200b\u200bthis, as mathematics say, degenerate The parallelogram is zero. It follows from the formula - sinus zero or 180 degrees is zero, and therefore the area is zero

Thus, if . Strictly speaking, the very vector product is zero vector, but in practice it is often neglected and written that it is simply zero.

Private case - vector product vector on himself:

With the help of a vector product, the collinearity of three-dimensional vectors can be checked, and we will also look at this task among others.

To solve practical examples may require trigonometric TableTo find it the values \u200b\u200bof sinuses.

Well, ignite the fire:

Example 1.

a) find the length of the vector art vectors if

b) find the square of the parallelogram built in the versions if

Decision: No, this is not a typo, the initial data in the clause conditions I intentionally made the same. Because making decisions will be different!

a) under the condition you need to find length Vector (vector art). According to the corresponding formula:

Answer:

Kohl soon asked about length, then in response, indicate the dimension - units.

b) under the condition required to find area A parallelogram built in vectors. The area of \u200b\u200bthis parallelogram is numerically equal to the length of the vector product:

Answer:

Please note that in response about the vector product of speech does not go at all, we were asked about square of Figure, accordingly, the dimension is square units.

We always look at what is required by condition, and, based on this, we formulate clear answer. It may seem keying, but there are enough keystones among teachers, and the task with good chances will return to refinement. Although this is not a particularly stretched quarid - if the answer is incorrect, then it seems that a person does not understand simple things and / or not an in the essence of the task. This moment should always be kept on control, solving any task in higher mathematics, and in other subjects too.

Where did the big bucchka "En"? In principle, it could additionally join the solution, but in order to reduce the record, I did not. I hope everyone understands that this is the designation of the same.

Popular example for self solutions:

Example 2.

Find a triangle area built in vectors if

The formula for finding the triangle area through the vector art is given in the comments to the definition. Solution and answer at the end of the lesson.

In practice, the task is really very common, triangles generally can torture.

To solve other tasks, we will need:

Properties of vector artwork

Some properties of vector work we have already considered, however, I will include them in this list.

For arbitrary vectors and arbitrary numbers, the following properties are fair:

1) In other sources of information, this item is usually not identified in the properties, but it is very important in practical terms. Therefore, let it be.

2) - the property is also disassembled above, sometimes it is called anti-commutative. In other words, the order of vectors matters.

3) - bleak or associative Laws of vector work. Constants are temporarily taken out of the vector work. Indeed, what do they do there?

4) - distributive or distribut Laws of vector work. With the disclosure of the brackets, there are no problems.

As a demonstration, consider a short example:

Example 3.

Find if

Decision: By condition, it is necessary to find the length of the vector product again. We bring our miniature:

(1) According to associative laws, we endure the constants for the redistribution of vector work.

(2) We endure the constant outside the module, while the module "eats" a "minus" sign. The length cannot be negative.

(3) Further is understandable.

Answer:

It's time to throw firewood into the fire:

Example 4.

Calculate the triangle area built in the vectors, if

Decision: Triangle Square Find the Formula . The snag is that the "CE" and "DE" vectors themselves are represented as sums of vectors. Algorithm here is standard and something resembles examples number 3 and 4 lessons Scalar product vectors. The solution for clarity to break into three stages:

1) In the first step, we express a vector product through a vector art, in fact, express vector via vector. About lengths not a word!

(1) We substitute the expression of vectors.

(2) Using distributional laws, reveal brackets according to the rule of multiplication of polynomials.

(3) Using associative laws, we endure all the constants beyond the vector works. Under the Malomal experience, 2 and 3 can be performed simultaneously.

(4) The first and last term is zero (zero vector) thanks to a pleasant property. In the second term, we use the anti-commutativeness property of the vector work:

(5) We give such components.

As a result, the vector turned out to be expressed through the vector, which was required to be achieved:

2) At the second step, we will find the length of the vector product you need. This action resembles Example 3:

3) Find the area of \u200b\u200bthe desired triangle:

Stages 2-3 solutions could be arranged with one line.

Answer:

The considered task is sufficiently disseminated in the tests, here is an example for an independent decision:

Example 5.

Find if

A brief solution and answer at the end of the lesson. Let's see how attentive you are when studying previous examples ;-)

Vector artwork of vectors in coordinates

defined in the orthonormal basis formula is expressed:

Formula and True Sprydskaya: In the upper line of the determinant, we write down the coordinate vectors, in the second and third lines "put" the coordinates of the vectors, and fit in strict order - First, the coordinates of the vector "VE", then the coordinates of the vector "Dubl-WE". If the vectors need to multiply in a different order, then the rows should be swapped in places:

Example 10.

Check whether the collinear will be the following space vectors:
but)
b)

Decision: Check is based on one of the statements of this lesson: if the collinear vectors, then their vector product is zero (zero vector): .

a) Welcome a vector art:

Thus, the vectors are not collinear.

b) Find a vector art:

Answer: a) not collinear, b)

This is perhaps all the basic information about the vector product of vectors.

This section will not be very large, since the tasks where the mixed product of vectors is used, a little. In fact, everything will be restricted into definition, geometric meaning and a couple of working formulas.

Mixed artwork of vectors is a work of three vectors.:

That's how they were lined up by a train and wait, would not wait when they were calculated.

First, again definition and picture:

Definition: Mixed work noncomplenar vectors, taken in this order, called the volume of parallelepipeda, built on the data of the vector, equipped with the "+" sign, if the basis is the right, and the sign "-", if the basis is left.

Perform a picture. Invisible lines are battered by the dotted line:

Immerse yourself in definition:

2) vectors taken in a certain order, that is, the rearrangement of vectors in the work, as you guess, does not pass without consequences.

3) Before commenting geometric meaning, I will note the obvious fact: mixed vectors is a number:. In the educational literature, the design may be somewhat different, I used to sign a mixed product through, and the result of calculations of the letter "PE".

A-priory mixed work is a parallelepiped volumebuilt in vectors (the figure is cleaned with red vectors and black lines). That is, the number is equal to the volume of this parallelepiped.

Note : The drawing is schematic.

4) Let's not re-steam with the concept of orientation of the basis and space. The meaning of the final part is that a minus sign can be added to the volume. Simple words, a mixed product may be negative :.

Directly from the definition follows the formula for calculating the volume of parallelepiped, built in vectors.

The area of \u200b\u200bthe parallelogram constructed in the versions is equal to the product of the lengths of these vectors at the angle of the angle, which lies between them.

Well, when the lengths of these vectors are given by conditions. However, it also happens that to apply the parallelogram area formula, built in vectors only after calculations by coordinates.
If lucky, and under the conditions are given the length of the vectors, then you just need to apply the formula that we have previously disassembled in the article. The area will be equal to the product of the modules on the sine corner between them:

Consider an example of calculating the area of \u200b\u200bthe parallelogram built in vectors.

A task: Pollogram built in vectors and. Find the area, if, and the angle between them is 30 °.
Express the vector through their values:

Perhaps you have a question - where did the zero come from? It is worth remembering that we work with vectors, and for them . Also note that if as a result we get expression, it will be converted to. Now we carry out the final calculations:

Let's return to the problem when the lengths of the vectors are not specified in conditions. If your parallelogram lies in the Cartesian coordinate system, it will be necessary to do the following.

Calculation of the side lengths of the figure defined by coordinates

To begin with, we find the coordinates of the vectors and take the corresponding coordinates of the beginning from the coordinates of the end. Let us assume the coordinates of the vector A (x1; y1; z1), and the vector B (x3; y3; z3).
Now we find the length of each vector. To do this, each coordinate must be elevated to the square, then fold the results obtained and from the final number to extract the root. According to our vectors there will be the following calculations:


Now it will be necessary to find a scalar product of our vectors. For this, their respective coordinates multiply and develop.

Having the lengths of the vectors and their scalar product, we can find the cosine of the angle lying between them .
Now we can find the sinus of the same angle:
Now we have all the necessary quantities, and we can easily find the area of \u200b\u200bthe parallelogram built in the versions of the already known formula.

Area parallelogrambuilt by vectorsIt is calculated as a product of the lengths of these vectors on the corner sinus between them. If only the coordinates of the vectors are known, the coordinate methods should be used to calculate the coordinate methods, including to determine the angle between vectors.

You will need

• - vector notion;
• - properties of vectors;
• - Cartesian coordinates;
• - trigonometric functions.

Instruction

• In the event that the lengths of the vectors and the angle between them are known, in order to find the area parallelogrambuilt by vectors, Find the product of their modules (vector lengths), on the sine of the angle between them s \u003d │a │ │ b │ sin (α).
• If the vectors are specified in the Cartesian coordinate system, then in order to find the area parallelogramBuilt on them, do the following:
• Find the coordinates of the vectors if they are not immediately given by the corresponding coordinates of the ends of the vectors, the coordinates from the beginning. For example, if the coordinates of the initial point of the vector (1; -3; 2), and the ultimate (2; -4; -5), then the coordinates of the vector will be (2-1; -4 + 3; -5-2) \u003d (1 ; -1; -7). Let the coordinates of the vector A (x1; y1; z1), vector B (x2; y2; z2).
• Find the lengths of each of the vectors. Take each of the coordinates of the vectors in the square, find their sum X1² + Y1² + Z1². From the resulting result, remove the square root. For the second vector, do the same procedure. Thus, it turns out │a│ and │ b.
• Find a scalar product of vectors. To do this, multiply their respective coordinates and fold the works │a b│ \u003d x1 x2 + y1 y2 + z1 z2.
• Determine the cosine of the angle between them for which the scalar product of the vectors, obtained in paragraph 3, divide the lengths of the vectors that were calculated in paragraph 2 (COS (α) \u003d │a b│ (│a│ │ b│)).
• The sinus of the resulting angle will be equal to the root square from the difference of the number 1, and the square of the cosine of the same angle calculated in clause 4 (1-cosqm (α)).
• Calculate Square parallelogrambuilt by vectors Having found the product of their lengths calculated in paragraph 2, and the result multiply the number after the calculations in P.5.
• In the event that the coordinates of the vectors are predetermined on the plane, when calculating the z coordinate is simply discarded. This calculation is a numerical expression of a vector product of two vectors.