Rule of vector practice. Vector vitvir vector_v
Vector vitvir- tse pseudovector, perpendicular to the plane, pobudovanoї on two spіvmultipliers, which is the result of the binary operation "vector multiplication" over vectors in the trivial Euclidean space. Vector tvir does not have the power of commutativity and associativity (є anticommutative) i, on vіdminu vіd scalar creation of vectorіv, є vector. Widely distinguished in rich technical and physical additions. For example, momentum and Lorentz force are mathematically written as a vector product. The vector extension of the corisny for "reversing" the perpendicularity of vectors is the module of the vector creation of two vectors of the additional extension of their modules, as they are perpendicular, and changes to zero, as the vectors are parallel or anti-parallel.
Significance vector witwear it is possible in a different way, and theoretically, in the space, whether there is a wideness of n, you can count additional n-1 vectors, taking away from your single vector, perpendicular to them all. But if tvir is surrounded by non-trivial binary creations with vector results, then the traditional vector tvir is assigned only to the trivial and seven-world spaces. The result of a vector creation, like a scalar one, lies in the Euclidean space metric.
On the other hand, the formula for calculating the coordinates of the vector scalar object in a three-dimensional rectangular coordinate system;
Appointment:
The vector complement of the vector a to the vector b in the space R 3 is called the vector c
The length of the vector c
|c|=|a||b|sin φ;
vector c orthogonal to skin vector s a and b;
the vector c of rectifications so that the trinity of vectors in abc is right;
the space R7 needs the associativity of the trio of vectors a, b, c.
Designation:
c===a×b
Rice. 1. The area of the parallelogram is equal to the module of the vector creation
Geometric power of vector art:
Necessary and sufficient mental colinearity of two non-zero vectors is equalness of zero to their vector creation.
Vector creative module dorivnyuє area S parallelogram inspired by vectors reduced to the cob aі b(Div. fig. 1).
Yakscho e- single vector, orthogonal vector aі b and vibranium so that three a,b,e- rights, and S- the area of the parallelogram induced on them (pointing to the cob), then the following formula is valid for the vector creation:
=S e
Fig.2. The volume of the parallelepiped with the variation of vector and scalar creation of vectors; dotted lines show the projections of the vector c on a × b and the vector a on b × c, the first line is the significance of scalar creations
Yakscho c- which vector, π
- be-yak flat, scho vengeance tsey vector, e- a single vector that lies near the plane π
i orthogonal to c,g- single vector orthogonal to the plane π
and straightening so that three vectors ecgє right, then for someone who lies at the square π
vector a the correct formula is:
=Pr e a |c|g
de Pr e a is the projection of the vector e onto a
|c|-modulus of the vector h
When choosing a vector and a scalar creation, you can use the parallelepiped, inspired by the vectors reduced to the cob a, bі c. So tvir three vectors are called zmishanim.
V=|a (b×c)|
The little one shows how this can be done in two ways: the geometric result is saved when replacing the “scalar” and “vector” creations:
V=a×b c=a b×c
The magnitude of the vector creation to lie in the sine of the cut between the primary vectors, then the vector tvir can be taken as the steps of perpendicularity of the vector in the same way, like the scalar tvir can be seen as the steps of parallelism. Vectorial addition of two single vectors to 1 (single vector), as well as vectors and perpendicular, and as 0 (zero vector), as vectors and parallel or anti-parallel.
Viraz for vector creation in Cartesian coordinates
Yakscho two vectors aі b assigned by their rectangular Cartesian coordinates, or rather, represented in an orthonormal basis
a = (a x, a y, a z)
b = (b x, b y, b z)
and the coordinate system is right, then your vector tvir may look
=(a y b z -a z b y ,a z b x -a x b z ,a x b y -a y b x)
For memorization ts_єї formulas:
i = ∑ε ijk a j b k
de ε ijk- a symbol of Levi-Chiviti.
The power of the scalar creation
Scalar tvіr vectorіv, vznachennya, dominion
Linear operations on vectors.
Vectors, basic concepts, designations, linear operations on them
A pair of її points is called a vector on the plane, in which case the first point is called an ear, and the other end - a vector.
Two vectors are called equal, because the stench is equal and co-directed.
Vectors that lie on one straight line are called co-directional because they are co-directed with one and the same vector that does not lie on this straight line.
Vectors that lie on one straight line or on parallel lines are called collinear, and collinear vectors that are not co-directed are called opposite-straight.
Vectors that lie on perpendicular lines are called orthogonal.
Appointment 5.4. sumoyu a+b vector_v a і b called a vector, which goes on the cob of a vector but at the end of the vector b , just like the cob vector b zbіgaєtsya with the end of the vector but .
Appointment 5.5. Retail a - b vector_v but і b such a vector is called h , which is the sum of the vector b yes vector but .
Appointment 5.6. Tvoromk a vector but per number k called vector b , collinear vector but , what is the module, equal | k||a |, that straight, scho zbіgaєtsya s straight | but at k>0 i length but at k<0.
The power of multiplying a vector by a number:
Power 1. k(a+b ) = k a+ k b.
Power 2. (k+m)a = k a+ m a.
Power 3. k(m a) = (km)a .
Last. Like non-zero vectors but і b kolіnearnі, then іsnuє such kolіkіst k, what b= k a.
The scalar product of two non-zero vectors aі b the number (scalar) is called, which allows for the addition of two vectors by the cosine of the cut φ between them. The scalar tvir can be designated in different ways, for example, like ab, a · b, (a , b), (a · b). In this order, the scalar tvir is good:
a · b = |a| · | b| cos φ
If we want one of the vectors to reach zero, then the scalar addition to zero.
Power of permutation: a · b = b · a(The type of permutation of multipliers in scalar twir does not change);
· The power of rozpodіlu: a · ( b · c) = (a · b) · c(The result does not lie in the order of multiplication);
Power of the day (hundreds of scalar multiplier): (λ a) · b = λ ( a · b).
Power of orthogonality (perpendicularity): as a vector aі b non-zero, their scalar tvir is equal to zero, only if q vectors are orthogonal (perpendicular one to one) a ┴ b;
Power of the square: a · a = a 2 = |a| 2 (the scalar addition of the vector itself is equal to the square of the th module);
How to coordinate vectors a=(x 1 , y 1 , z 1 ) b=(x 2 , y 2 , z 2 ), then the scalar solid is one a · b= x 1 x 2 + y 1 y 2 + z 1 z 2.
Vector vector conduction. Appointment: Under the vector creation of two vectors and a vector is understood, for which:
The module of the additional area of the parallelogram, inspired by these vectors, tobto. , de cut between vectors ta
Tsey vector of perpendicular vectors, which are multiplied, tobto.
As vectors are not collinear, the stench satisfies the right of the trinity of vectors.
The power of the vector creation:
1.When changing the order of the multipliers of the vector TV, you change your own sign of the return, saving the module, tobto.
2 .Vector square is equal to zero-vector, tobto.
3 .The scalar multiplier can be blamed for the symbol of the vector create, tobto.
4 .For any three vectors, equality is fair
5 .Necessary and sufficient mind collinarity of two vectors and:
Kut mizh vectors
So that we could introduce the concept of the vector creation of two vectors, it is necessary to sort out such concepts, as a way to cut between these vectors.
Come on, we are given two vectors $\overline(α)$ and $\overline(β)$. Let's take the point $O$ in space and add the vector $\overline(α)=\overline(OA)$ i $\overline(β)=\overline(OB)$, then $AOB$ will be called the cut between with vectors (Fig. 1).
Signature: $∠(\overline(α),\overline(β))$
Understanding the vector creative vector
Appointment 1
A vector created by two vectors is a vector that is perpendicular to both given vectors, and the second vector is a more efficient addition of the two vectors with the sine of the kuta between the given vectors, and also the vector of the two cobs has the same orientation, like the Cartesian coordinate system.
Significant: $\overline(α)х\overline(β)$.
Mathematically, it looks like this:
- $|\overline(α)x\overline(β)|=|\overline(α)||\overline(β)|sin∠(\overline(α),\overline(β))$
- $\overline(α)x\overline(β)⊥\overline(α)$, $\overline(α)x\overline(β)⊥\overline(β)$
- $(\overline(α)x\overline(β),\overline(α),\overline(β))$ i $(\overline(i),\overline(j),\overline(k))$ however orientation (Fig. 2)
It is obvious that the current tvir vector is equal to the zero vector in two directions:
- How long is one or both vectors equal to zero.
- How to cut between these two vectors equal to $180^\circ$ or $0^\circ$ (scales in which direction sine is equal to zero).
Sob in the first place, how to know vector tvir vektorіv, look at the point below, apply the solution.
butt 1
Find the value of the vector $\overline(δ)$, which will be the result of the vector creation of vectors, with coordinates $\overline(α)=(0,4,0)$ and $\overline(β)=(3,0,0 ) $.
Solution.
Let us visualize q vectors and y in the Cartesian coordinate space (Fig. 3):
Figure 3. Vectors in the Cartesian coordinate space. Author24 - Internet exchange of student works
Bachimo, that qi vectors lie on the axes $Ox$ and $Oy$ clearly. Otzhe, kut mіzh them dovnyuvatime $90^\circ$. We know about these vectors:
$|\overline(α)|=\sqrt(0+16+0)=4$
$|\overline(β)|=\sqrt(9+0+0)=3$
Then, for assignment 1, we take the module $|\overline(δ)|$
$|\overline(δ)|=|\overline(α)||\overline(β)|sin90^\circ=4\cdot 3\cdot 1=12$
Suggestion: $12$.
Calculation of the vector creation for the coordinates of the vectors
Z vyznachennya 1 vіdrazu vіplyvaє sposіb znakhodzhennya vector creation for two vectorіv. Oskіlki vector, krіm znachlennya, maє shche th directly, it is impossible to know it only for an additional scalar quantity. Ale krіm new іsnuіє sposіb znakhodzhennya for additional coordinates given to us vectorіv.
Let us give us vectors $\overline(α)$ i $\overline(β)$, so that we can calculate the coordinates $(α_1,α_2,α_3)$ i $(β_1,β_2,β_3)$, obviously. The same vector of the vector creation (and its own coordinates) can be known by the following formula:
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\α_1&α_2&α_3\\β_1&β_2&β_3\end(vmatrix)$
Otherwise, rozkrivayuchi vyznachnik, take such coordinates
$\overline(α)х\overline(β)=(α_2 β_3-α_3 β_2,α_3 β_1-α_1 β_3,α_1 β_2-α_2 β_1)$
butt 2
Find the vector of the vector creation of collinear vectors $\overline(α)$ and $\overline(β)$ with coordinates $(0,3,3)$ and $(-1,2,6)$.
Solution.
Speeding up with a formula that has been induced higher. Take away
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\0&3&3\-1&2&6\end(vmatrix)=(18 - 6)\overline(i)-(0+3)\overline(j)+(0+3)\overline(k)=12\overline(i)-3\overline(j)+3\overline(k) ) = (12,-3,3) $
Value: $ (12,-3.3) $.
The power of the vector creative vector
For more than three shifts in $\overline(α)$, $\overline(β)$ і $\overline(γ)$, and also $r∈R$, the advancing power is fair:
butt 3
Find the area of the parallelogram, the vertices of which are coordinates $(3,0,0)$, $(0,0,0)$, $(0,8,0)$ and $(3,8,0)$.
Solution.
The back of the head is represented by a parallelogram at the coordinate space (Fig. 5):
Figure 5. Parallelogram at the coordinate space. Author24 - Internet exchange of student works
Bachimo, that the two sides of this parallelogram were inspired by additional collinear vectors with coordinates $\overline(α)=(3,0,0)$ and $\overline(β)=(0,8,0)$. Vikoristovuyuchi fourth power, otrimaemo:
$S=|\overline(α)x\overline(β)|$
We know the vector $\overline(α)х\overline(β)$:
$\overline(α)x\overline(β)=\begin(vmatrix)\overline(i)&\overline(j)&\overline(k)\\3&0&0\\0&8&0\end(vmatrix)=0\overline (i)-0\overline(j)+24\overline(k)=(0,0,24)$
Otzhe
$S=|\overline(α)x\overline(β)|=\sqrt(0+0+24^2)=24$
ZMISHANY VIROB TROCH VECTORIV AND YOGO POWER
Zmіshanim creative three vectors name the number that is good. be appointed 
. Here the first two vectors are multiplied vectorially and then the subtracting vector is scalarly multiplied by the third vector . Obviously, such a TV is a sprat.
Let's look at the power of the mixed creation.
- geometric sense crazy creation. Zmіshane tvir 3 vectors with accuracy up to the sign of the agreement of the parallelepiped, induced by these vectors, like at the ribs, tobto. .
In such a manner,
.
proof. Vіdklademo vektori vіd zagalnogo cob and pobuduєmo on them paralepiped. Significantly and respectfully, scho. For the purpose of the scalar creation
Allowing what i know through h the height of the parallelepiped, we know.
In such a manner, at
Well, then th. Father, .
Ob'ednuyuchi insults and vipadki, otrimuєmo either.
Z confirmation of the quality of the zokrem is viplivay, that the third vector is right, then zmishane tvir, and yakshcho - leva, then.
- For whatever vectors , , equality is fair
The proof of the authority's power is evident from the authority 1. True, it is easy to show that. Until then, the signs "+" and "-" are taken at the same time, because kuti mizh vectors ta і one hour gostrі or stupid.
- When rearranging, whether there are two spіvmulnіnіv zmіshanі tvіr change the sign.
It’s true, as if we can look at the confusion of TV, then, for example, or
- Zmіshany tvіr tіlki tіlki tіlki і, if іz сpіvmіnnіkіv dоrіvnyuє zero аbо vectors аrе coplanar.
proof.
Including, the necessary and sufficient mental coplanarity of 3 vectors and the equality to zero of their mixed creation. In addition, it is obvious that three vectors establish a basis for space, for example.
As well as the vectors and tasks in the coordinate form, it is possible to show that these changes are known by the formula:
.Thus, zmіshane tvіr dоrіvnyuє vyznachnik of the third order, which has the coordinates of the first vector in the first row, the coordinates of another vector in the other row, and the coordinates of the third vector in the third row.
apply.
ANALYTICAL GEOMETRY IN SPACE
Rivnyannia F(x, y, z)= 0 is assigned to the space Oxyz deaku surface, tobto. geometrical place point, coordinates of which x, y, z satisfy whomever is jealous. The line is called equal to the surface, and x, y, z- current coordinates.
However, often the surface is asked not by equals, but as an impersonal point of space, which may have that other power. And here it is necessary to know the equivalence of the surface, from її geometrical powers.
AREA.
NORMAL AREA VECTOR.
LEVELING THE PLANE TO PASS THROUGH A GIVEN POINT
Let's look at the expanse of a large area σ. The position is dependent on the given vector perpendicular to the given plane, that fixed point M0(x0, y 0, z0) that lies near the plane σ.
The vector perpendicular to the plane σ is called normal vector qієї area. Let the vector have coordinates.
We can see that the plane σ is equal to pass through the qi point M0 and may be a normal vector. For which we take on the plane σ a sufficient point M(x, y, z) i look at the vector.
For whatever point MО σ vector Tsya jealousy is the mind of the point MО σ. It is fair for all points in the plane and breaks down, like only a point M lean back pose with a plane σ.
How to know through the radius-vector of a point M, is the radius vector of the point M0, then th equal can be recorded at a glance
Tse equal is called vector equal to the area. Let's write yoga in the coordinate form. Oscilki, then
Otzhe, we took away the flatness of the area, to pass this point. In this way, in order to fold the flatness of the plane, it is necessary to know the coordinates of the normal vector and the coordinates of the deuce point that lie on the plane.
Respectfully, that the plane is equal to the equal of the 1st stage of the flow coordinates x, yі z.
apply.
ZAGALNE RIVNYANNYA PLANE
Can you show how equal the first step is to Cartesian coordinates x, y, zє equal to deykoї area. The price is recorded as:
Ax+By+Cz+D=0
and is called wild jealous planes, and the coordinates A, B, C here are the coordinates of the normal vector of the area.
Let's look at the surroundings of the outrageous jealousy. Of course, as the plane of the coordinate system is expanded, it means that one or the other coefficients of alignment are adjusted to zero.
A - the core of the vіdrіzka, which is seen by the plane on the axis Ox. Similarly, it can be shown that bі c- Dovzhini vіdrіzkіv, scho vіdsіkayutsya flat on the axes, scho to be seen. Ouchі Oz.
Rivnyannyam flats at the windbreaks are handy to scorch for raising the flats.
