The main property of direction cosines. Formula for calculating direction cosines of a vector for spatial problems Coordinates modulus and direction cosines of a vector
Let the vector be given. Unit vector in the same direction as 
(unit vector 
) is found by the formula:
.
Let the axis 
forms angles with the coordinate axes 
.Direction cosines of the axis 
The cosines of these angles are called:. If the direction 
given by a unit vector 
, then the direction cosines serve as its coordinates, i.e.:
.
The direction cosines are related to each other by the relation:
If the direction 
given by an arbitrary vector 
, then find the unit vector of this vector and, comparing it with the expression for the unit vector 
, get:
Scalar product
Dot product
two vectors 
And 
is a number equal to the product of their lengths and the cosine of the angle between them: 
.
The scalar product has the following properties:
Hence, 
.
Geometric meaning of the dot product: scalar product of a vector and a unit vector 
equal to the projection of the vector 
to the direction determined 
, i.e. 
.
The following table of multiplication of unit vectors follows from the definition of the scalar product: 
:
.
If vectors are given by their coordinates 
And 
, i.e. 
,
, then, multiplying these vectors scalarly and using the multiplication table of unit vectors, we obtain the expression for the scalar product 
through vector coordinates:
.
Vector artwork
Cross product of a vector
to vector 
called a vector 
, the length and direction of which are determined by the conditions:
The vector product has the following properties:
From the first three properties it follows that the vector multiplication of a sum of vectors by a sum of vectors obeys the usual rules for multiplying polynomials. You just need to make sure that the order of the factors does not change.
The basic vectors are multiplied as follows:
If 
And 
, then taking into account the properties of the vector product of vectors, we can derive a rule for calculating the coordinates of the vector product from the coordinates of the factor vectors:
If we take into account the above rules for multiplying unit vectors, then:
A more compact form of writing an expression for calculating the coordinates of the vector product of two vectors can be constructed by introducing the concept of a determinant of a matrix.
Let us consider the special case when the vectors 
And 
belong to the plane 
, i.e. they can be represented as 
And 
.
If the coordinates of the vectors are written in table form as follows: 
, then we can say that a square matrix of the second order is formed from them, i.e. size 
, consisting of two rows and two columns. Each square matrix is associated with a number, which is calculated from the elements of the matrix according to certain rules and is called a determinant. The determinant of a second-order matrix is equal to the difference between the products of the elements of the main diagonal and the secondary diagonal:
.
In this case:
The absolute value of the determinant is thus equal to the area of the parallelogram constructed on the vectors 
And 
, both on the sides.
If we compare this expression with the vector product formula (4.7), then:
|
|
This expression is a formula for calculating the determinant of a third-order matrix from the first row.
Thus:
Determinant of a third-order matrix is calculated as follows:
and is the algebraic sum of six terms.
The formula for calculating the determinant of a third-order matrix is easy to remember if you use ruleSarrus, which is formulated as follows:
Each term is the product of three elements located in different columns and different rows of the matrix;
The products of elements forming triangles with a side parallel to the main diagonal have a plus sign;
The products of elements belonging to the secondary diagonal and two products of elements forming triangles with a side parallel to the secondary diagonal have a minus sign.
The sum of the squares of the direction cosines is equal to one.
If the direction cosines of the vector are known, then its coordinates can be found using the formulas: Similar formulas apply in the three-dimensional case - if the direction cosines of the vector are known, then its coordinates can be found using the formulas:
9 Linear dependence and linear independence of vectors. Basis on the plane and in space
A set of vectors is called system of vectors.
linearly dependent, if there are numbers that are not all equal to zero at the same time, that
A system of vectors is called linearly independent, if equality is possible only for , i.e. when the linear combination on the left side of the equality is trivial.
1. One vector also forms a system: at - linearly dependent, and at - linearly independent.
2. Any part of a system of vectors is called subsystem.
1. If a system of vectors includes a zero vector, then it is linearly dependent
2. If a system of vectors has two equal vectors, then it is linearly dependent.
3. If a system of vectors has two proportional vectors, then it is linearly dependent.
4. A system of vectors is linearly dependent if and only if at least one of the vectors is a linear combination of the others.
5. Any vectors included in a linearly independent system form a linearly independent subsystem.
6. A system of vectors containing a linearly dependent subsystem is linearly dependent.
7. If a system of vectors is linearly independent, and after adding a vector to it it turns out to be linearly dependent, then the vector can be expanded into vectors and, moreover, in a unique way, i.e. the expansion coefficients can be found uniquely.
Basis on a plane and in space is called a maximal system of vectors that is linearly independent on a plane or in space (adding another vector to the system makes it linearly dependent).
Thus, a basis on a plane is any two non-collinear vectors taken in a certain order, and a basis in space is any three non-coplanar vectors taken in a certain order.
Let be a basis in space, then, according to T. 3, any vector of space can be decomposed in a unique way into basis vectors: . The expansion coefficients are called the coordinates of the vector in the basis
Writing linear operations on vectors through coordinates:
a) addition and subtraction: - basis
b) multiplication by the number R:
The formulas follow from the properties of linear operations.
10 Coordinates of the vector relative to the basis. Orty
Basis in free vector space V 3 is any ordered triple of non-coplanar vectors.
Let IN :a 1,a 2,a 3– fixed basis in V 3.
Coordinates vector b relative to the basis IN called an ordered triple of numbers ( x, y, z), incl. b=x· a 1 +ya 2 +z· a 3.
Designation:b={x, y, z} B Note: The coordinates of a fixed vector mean the coordinates of the corresponding free vector.
Theorem1: The correspondence between V 3 and R 3 for a fixed basis is one-to-one, i.e. b V 3 ! {x, y, z) R 3 and ( x, y, z) R 3 ! b V 3, incl. b={x, y, z} B
The correspondence between a vector and its coordinates in a given basis has the following properties:
1. Let b 1 ={x 1 , y 1 , z 1} B , b 2 ={x 2 , y 2 , z 2} B b 1 + b 2 ={x 1 + x 2 , y 1 + y 2 , z 1 + z 2} B
2. Let b={x, y, z} B , λR λ b={ λ· x, λ· y, λ· z} B
3. Let b 1 || b 2 , b 1 = {x 1 , y 1 , z 1} B
, b 2 ={x 2 , y 2 , z 2} B
(Here: any number).
Unit vector, directed along the X axis, is denoted i, unit vector, directed along the Y axis, is denoted j, A unit vector, directed along the Z axis, is denoted k. Vectors i, j, k are called orts– they have single modules, that is
i = 1, j = 1, k = 1
11 scalar product of vectors. Angle between vectors. Condition for vector orthogonality
This is a number equal to the product of the lengths of these vectors and the cosine of the angle between them.
Dot product of vectors in terms of their coordinates
Dot product of vectors X, Y, Z and :
where is the angle between the vectors and ; if either, then
From the definition of the scalar product it follows that where, for example, is the magnitude of the projection of the vector onto the direction of the vector.
Scalar squared vector:
Properties of the dot product:
Angle between vectors
Conditions for vector orthogonality.
Two vector a and b orthogonal (perpendicular), if their scalar product is equal to zero a· b= 0
So in the case of a plane vector problem
a= (a x ;a y )and b= (b x ;b y )
orthogonal ifa b= a x b x + a y b y = 0
12 vector product of vectors, its properties. Condition for collinearity of vectors
The cross product of a vector and a vector is a vector denoted by a symbol and defined by the following three conditions:
1). The modulus of the vector is equal to , where is the angle between the vectors and ;
2). The vector is perpendicular to each of the vector and ;
3). The direction of the vector corresponds to the “right hand rule”. This means that if the vectors , and are brought to a common origin, then the vector should be directed in the same way as the middle finger of the right hand, the thumb of which is directed along the first factor (that is, along the vector), and the index finger - along the second (that is, along vector). The vector product depends on the order of the factors, namely: .
The modulus of the vector product is equal to the area S of a parallelogram constructed on the vectors and : .
The vector product itself can be expressed by the formula,
where is the unit vector of the vector product.
The cross product vanishes if and only if the vectors and are collinear. In particular, .
If the coordinate axes system is right and the vectors and are specified in this system by their coordinates:
then the vector product of a vector and a vector is determined by the formula
A vector is collinear to a nonzero vector if and only if the coordinates
vectors are proportional to the corresponding coordinates of the vector, i.e.
Linear operations on vectors specified by their coordinates in space are performed in a similar way.
13 mixed product of vectors. Its properties. Condition for coplanarity of vectors
Mixed product of three vectors, , is a number equal to the scalar product of a vector and a vector:
Properties of a mixed product:
3° Three vectors are coplanar if and only if
4° A triple of vectors is right if and only if . If , then the vectors , and form the left triplet of vectors.
10° Jacobi identity:
If the vectors , and are given by their coordinates, then their mixed product is calculated using the formula
Vectors parallel to one plane or lying on the same plane are called coplanar vectors.
Conditions for coplanarity of vectors
Three vectors are coplanar if their mixed product is zero.
Three vectors are coplanar if they are linearly dependent.
15 different types of line and plane equations
Any straight line on the plane can be specified by a first-order equation
Ax + Wu + C = 0,
Moreover, the constants A and B are not equal to zero at the same time. This first order equation is called general equation of a straight line. Depending on the values of constants A, B and C, the following special cases are possible:
C = 0, A ≠0, B ≠ 0 – the straight line passes through the origin
A = 0, B ≠0, C ≠0 (By + C = 0) - straight line parallel to the Ox axis
B = 0, A ≠0, C ≠ 0 (Ax + C = 0) – straight line parallel to the Oy axis
B = C = 0, A ≠0 – the straight line coincides with the Oy axis
A = C = 0, B ≠0 – the straight line coincides with the Ox axis
The equation of a straight line can be presented in different forms depending on any given initial conditions.
these are the cosines of the angles that the vector forms with the positive semi-axes of coordinates. Direction cosines uniquely specify the direction of the vector. If a vector has length 1, then its direction cosines are equal to its coordinates. In general, for a vector with coordinates ( a; b; c) direction cosines are equal:
where a, b, g are the angles made by the vector with the axes x, y, z respectively.
21) Decomposition of a vector in unit vectors. The unit vector of the coordinate axis is denoted by , the axes by , and the axes by (Fig. 1).
For any vector that lies in the plane, the following expansion takes place:
If the vector 
located in space, then the expansion in unit vectors of the coordinate axes has the form:
22)Dot product two non-zero vectors and the number equal to the product of the lengths of these vectors and the cosine of the angle between them is called:
23)Angle between two vectors
If the angle between two vectors is acute, then their scalar product is positive; if the angle between the vectors is obtuse, then the scalar product of these vectors is negative. The scalar product of two nonzero vectors is equal to zero if and only if these vectors are orthogonal.
24) The condition of parallelism and perpendicularity of two vectors.
Condition for vectors to be perpendicular
Vectors are perpendicular if and only if their scalar product is zero. Given two vectors a(xa;ya) and b(xb;yb). These vectors will be perpendicular if the expression xaxb + yayb = 0.
25) Vector product of two vectors.
The vector product of two non-collinear vectors is a vector c=a×b that satisfies the following conditions: 1) |c|=|a| |b| sin(a^b) 2) c⊥a, c⊥b 3) Vectors a, b, c form a right-hand triplet of vectors.
26) Collinear and coplanar vectors..
Vectors are collinear if the abscissa of the first vector is related to the abscissa of the second in the same way as the ordinate of the first is to the ordinate of the second. Given two vectors a (xa;ya) And b (xb;yb). These vectors are collinear if xa = x b And y a = y b, Where R.
Vectors −→ a,−→b and −→ c are called coplanar, if there is a plane to which they are parallel.
27) Mixed product of three vectors. Mixed product of vectors- scalar product of vector a and the vector product of vectors b and c. Find the mixed product of vectors a = (1; 2; 3), b = (1; 1; 1), c = (1; 2; 1).
Solution:
1·1·1 + 1·1·2 + 1·2·3 - 1·1·3 - 1·1·2 - 1·1·2 = 1 + 2 + 6 - 3 - 2 - 2 = 2
28) The distance between two points on a plane. The distance between two given points is equal to the square root of the sum of the squared differences of the same coordinates of these points.
29) Division of a segment in this relation. If point M(x; y) lies on a line passing through two given points ( , ) and ( , ), and a relation is given in which point M divides the segment , then the coordinates of point M are determined by the formulas
If point M is the midpoint of the segment, then its coordinates are determined by the formulas
30-31. Slope of a straight line is called the tangent of the angle of inclination of this line. The slope of a straight line is usually denoted by the letter k. Then by definition
Equation of a straight line with slope has the form where k- straight line slope, b– some real number. Using the equation of a straight line with an angle coefficient, you can specify any straight line that is not parallel to the axis Oy(for a straight line parallel to the ordinate axis, the angular coefficient is not defined).
33. General equation of a straight line on a plane. Equation of the form 
There is general equation of a line Oxy. Depending on the values of constants A, B and C, the following special cases are possible:
C = 0, A ≠0, B ≠ 0 – the straight line passes through the origin
A = 0, B ≠0, C ≠0 (By + C = 0) - straight line parallel to the Ox axis
B = 0, A ≠0, C ≠ 0 (Ax + C = 0) – straight line parallel to the Oy axis
B = C = 0, A ≠0 – the straight line coincides with the Oy axis
A = C = 0, B ≠0 – the straight line coincides with the Ox axis
34.Equation of a line in segments on a plane in a rectangular coordinate system Oxy has the form where a And b- some non-zero real numbers. This name is not accidental, since the absolute values of numbers A And b equal to the lengths of the segments that the straight line cuts off on the coordinate axes Ox And Oy respectively (segments are counted from the origin). Thus, the equation of a line in segments makes it easy to construct this line in a drawing. To do this, you should mark the points with coordinates and in a rectangular coordinate system on the plane, and use a ruler to connect them with a straight line.
35. The normal equation of a line has the form
where is the distance from the straight line to the origin; – the angle between the normal to the line and the axis.
The normal equation can be obtained from the general equation (1) by multiplying it by the normalizing factor, the sign is opposite to the sign so that .
The cosines of the angles between the straight line and the coordinate axes are called direction cosines, – the angle between the straight line and the axis, – between the straight line and the axis:
Thus, the normal equation can be written in the form
Distance from point to a straight line determined by the formula 
36. The distance between a point and a line is calculated using the following formula: 
where x 0 and y 0 are the coordinates of the point, and A, B and C are coefficients from the general equation of the line
37. Reducing the general equation of a line to normal. The equation and the plane in this context do not differ from each other in anything other than the number of terms in the equations and the dimension of space. Therefore, first I will say everything about the plane, and at the end I will make a reservation about the straight line.
Let the general equation of the plane be given: Ax + By + Cz + D = 0.
;. we get the system: g;Mc=cosb, MB=cosa Let's bring it to normal form. To do this, we multiply both sides of the equation by the normalizing factor M. We get: Max+Mvu+MCz+MD=0. In this case MA=cos;.g;Mc=cosb, MB=cosa we obtain the system:
M2 B2=cos2b
M2 C2=cos2g
Adding up all the equations of the system, we get M*(A2 +B2+C2)=1 Now all that remains is to express M from here in order to know by which normalizing factor the original general equation must be multiplied to bring it to normal form:
M=-+1/ROOT KV A2 +B2 +C2
MD must always be less than zero, therefore the sign of the number M is taken opposite to the sign of the number D.
With the equation of a straight line, everything is the same, only from the formula for M you should simply remove the term C2.
| Ax + By + Cz + D = 0, |
38.General equation of the plane in space is called an equation of the form
Where A 2 + B 2 + C 2 ≠ 0 .
In three-dimensional space in the Cartesian coordinate system, any plane is described by an equation of the 1st degree (linear equation). And conversely, any linear equation defines a plane.
40.Equation of a plane in segments. In a rectangular coordinate system Oxyz in three-dimensional space an equation of the form 
, Where a, b And c– non-zero real numbers are called equation of the plane in segments. Absolute values of numbers a, b And c equal to the lengths of the segments that the plane cuts off on the coordinate axes Ox, Oy And Oz respectively, counting from the origin. Sign of numbers a, b And c shows in which direction (positive or negative) the segments are plotted on the coordinate axes
41) Normal plane equation.
The normal equation of a plane is its equation written in the form
where , , are the direction cosines of the plane normal, e
p is the distance from the origin to the plane. When calculating the direction cosines of the normal, it should be assumed that it is directed from the origin to the plane (if the plane passes through the origin, then the choice of the positive direction of the normal is indifferent).
42) Distance from a point to a plane.Let the plane be given by the equation 
and a point is given. Then the distance from the point to the plane is determined by the formula
 |
Proof. The distance from a point to a plane is, by definition, the length of the perpendicular drawn from the point to the plane
Angle between planes
Let the planes and be specified by the equations and , respectively. You need to find the angle between these planes.
The planes, intersecting, form four dihedral angles: two obtuse and two acute or four right angles, and both obtuse angles are equal to each other, and both acute angles are also equal to each other. We will always look for an acute angle. To determine its value, we take a point on the line of intersection of the planes and at this point in each of
planes, we draw perpendiculars to the intersection line.
Def. 1.5.6. Direction cosines vector A let's call the cosines of the angles that this vector forms with the basis vectors, respectively, i , j , k .
Direction cosines of a vector A = (X, at, z) are found by the formulas:
The sum of the squares of the direction cosines is equal to one:
Direction cosines of a vector a
are the coordinates of its unit vector: .
Let the basis vectors i , j , k postponed from a common point ABOUT. We will assume that the orts specify the positive directions of the axes Oh, OU, Oz. Point set ABOUT (origin) and orthonormal basis i , j , k called Cartesian rectangular coordinate system in space. Let A– an arbitrary point in space. Vector A = OA= x i + y j + z k called radius vector points A, coordinates of this vector ( x, y, z) are also called point coordinates A(designation: A(x, y, z)). Coordinate axes Oh, OU, Oz also called, respectively, the axis abscissa, axis ordinate, axis applicate.
If a vector is given by the coordinates of its starting point IN 1 (x 1 , y 1 , z 1) and end point IN 2 (x 2 ,
y 2 , z 2), then the coordinates of the vector are equal to the difference between the coordinates of the end and the beginning: (since 
).
Cartesian rectangular coordinate systems on the plane and on the line are determined in exactly the same way with corresponding quantitative (in accordance with the dimension) changes.
Solving typical problems.
Example 1. Find the length and direction cosines of a vector A = 6i – 2j -3k .
Solution. Vector length: 
. Direction cosines: 
.
Example 2. Find vector coordinates A , forming equal acute angles with the coordinate axes, if the length of this vector is equal to .
Solution. Since , then substituting into formula (1.6), we get 
. Vector A
forms acute angles with the coordinate axes, so the ort 
. Therefore, we find the coordinates of the vector 
.
Example 3. Three non-coplanar vectors are given e 1 = 2i – k , e 2 = 3i + 3j , e 3 = 2i + 3k . Expand vector d = i + 5j - 2k by basis e 1 , e 2 , e 3 .
