Product of a vector and a number. What vector is called the sum of two vectors? Give the definition of the product of a vector and a number
Matrix of sizes m by n.
Matrix size m by n is a collection of mn real numbers or elements of another structure (polynomials, functions, etc.), written in the form of a rectangular table, which consists of m rows and n columns and taken in round or rectangular or double straight brackets. In this case, the numbers themselves are called matrix elements and each element is associated with two numbers - the row number and the column number. A matrix of size n by n is called square matrix of nth order, i.e. the number of rows is equal to the number of columns. Triangular - a square matrix in which all elements below or above the main diagonal are equal to zero. A square matrix is called diagonal , if all its off-diagonal elements are equal to zero. Scalar matrix - a diagonal matrix whose main diagonal elements are equal. A special case of a scalar matrix is the identity matrix. Diagonal a matrix in which all diagonal elements are equal to 1 is called single matrix and is denoted by the symbol I or E. A matrix whose elements are all zero is called null matrix and is denoted by the symbol O.
Multiplying matrix A by a number λ (symbol: λ A) consists in constructing a matrix B, the elements of which are obtained by multiplying each element of the matrix A by this number, that is, each element of the matrix B equals
Properties of multiplying matrices by a number
1. 1*A = A; 2. (Λβ)A = Λ(βA) 3. (Λ+β)A = ΛA + βA
4. Λ(A+B) = ΛA + ΛB
Matrix addition A + B is the operation of finding a matrix C, all elements of which are equal to the pairwise sum of all corresponding matrix elements A And B, that is, each element of the matrix C equals
Properties of matrix addition
5.commutativity) a+b=b+a
6.associativity.
7.addition with zero matrix;
8.existence of an opposite matrix (the same thing but there are minuses everywhere before each number)
Matrix multiplication - there is a matrix calculation operation C, the elements of which are equal to the sum of the products of the elements in the corresponding row of the first factor and column of the second.
Number of columns in the matrix A must match the number of rows in the matrix B. If matrix A has dimension, B- , then the dimension of their product AB = C There is .
Properties of matrix multiplication
1.associativity; (see above)
2. the product is not commutative;
3.the product is commutative in the case of multiplication with the identity matrix;
4.fairness of the distributive law; A*(B+C)=A*B+A*C.
5.(ΛA)B = Λ(AB) = A(ΛB);
2. Determinant of a square matrix of the first and nth order
The determinant of a matrix is a polynomial of the elements of a square matrix (that is, one in which the number of rows and columns is equal to
Determination via expansion in the first row
For a first order matrix determinant is the only element of this matrix itself:
For a matrix of determinants is defined as
For a matrix, the determinant is specified recursively:
, where is an additional minor to the element a 1j. This formula is called line expansion.
In particular, the formula for calculating the determinant of a matrix is:
= a 11 a 22 a 33 − a 11 a 23 a 32 − a 12 a 21 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 − a 13 a 22 a 31
Properties of determinants
When adding a linear combination of other rows (columns) to any row (column), the determinant does not change.
§ If two rows (columns) of a matrix coincide, then its determinant is equal to zero.
§ If two (or several) rows (columns) of a matrix are linearly dependent, then its determinant is equal to zero.
§ If you rearrange two rows (columns) of a matrix, then its determinant is multiplied by (-1).
§ The common factor of the elements of any series of a determinant can be taken out of the sign of the determinant.
§ If at least one row (column) of the matrix is zero, then the determinant is equal to zero.
§ The sum of the products of all elements of any row by their algebraic complements is equal to the determinant.
§ The sum of the products of all elements of any series by the algebraic complements of the corresponding elements of a parallel series is equal to zero.
§ The determinant of the product of square matrices of the same order is equal to the product of their determinants (see also the Binet-Cauchy formula).
§ Using index notation, the determinant of a 3x3 matrix can be defined using the Levi-Civita symbol from the relation:
Inverse matrix.
Inverse matrix - such a matrix A−1, when multiplied by which the original matrix A results in the identity matrix E:
Conditional existence:
A square matrix is invertible if and only if it is non-singular, that is, its determinant is not equal to zero. For non-square matrices and singular matrices, there are no inverse matrices.
Formula for finding
If the matrix is invertible, then to find the inverse matrix you can use one of the following methods:
a) Using a matrix of algebraic additions
C T- transposed matrix of algebraic additions;
The resulting matrix A−1 and will be the inverse. The complexity of the algorithm depends on the complexity of the algorithm for calculating the determinant O det and is equal to O(n²)·O det.
In other words, the inverse matrix is equal to one divided by the determinant of the original matrix and multiplied by the transposed matrix of algebraic additions (the minor is multiplied by (-1) to the power of the space it occupies) of the elements of the original matrix.
4. System of linear equations. System solution. Compatibility and incompatibility of the system. matrix method for solving a system of n linear equations with n variables. Krammer's theorem.
System m linear equations with n unknown(or, linear system) in linear algebra is a system of equations of the form
 | (1) |
Here x 1 , x 2 , …, x n- unknowns that need to be determined. a 11 , a 12 , …, a mn- system coefficients - and b 1 , b 2 , … b m- free members - are assumed to be known. Coefficient indices ( a ij) systems denote equation numbers ( i) and unknown ( j), at which this coefficient stands, respectively.
System (1) is called homogeneous, if all its free terms are equal to zero ( b 1 = b 2 = … = b m= 0), otherwise - heterogeneous.
System (1) is called square, if number m equations equal to the number n unknown.
Solution systems (1) - set n numbers c 1 , c 2 , …, c n, such that the substitution of each c i instead of x i into system (1) turns all its equations into identities.
System (1) is called joint, if it has at least one solution, and non-joint, if she does not have a single solution.
A joint system of type (1) may have one or more solutions.
Solutions c 1 (1) , c 2 (1) , …, c n(1) and c 1 (2) , c 2 (2) , …, c n(2) joint systems of the form (1) are called various, if at least one of the equalities is violated:
| c 1 (1) = c 1 (2) , c 2 (1) = c 2 (2) , …, c n (1) = c n (2) . |
Matrix form
A system of linear equations can be represented in matrix form as:
Ax = B.
If a column of free terms is added to matrix A on the right, then the resulting matrix is called extended.
Direct methods
Cramer's method (Cramer's rule)- a method for solving quadratic systems of linear algebraic equations with a non-zero determinant of the main matrix (and for such equations there is a unique solution). Named after Gabriel Cramer (1704–1752), who invented the method.
Description of the method
For system n linear equations with n unknown (over an arbitrary field)
with the determinant of the system matrix Δ different from zero, the solution is written in the form
(the i-th column of the system matrix is replaced by a column of free terms).
In another form, Cramer’s rule is formulated as follows: for any coefficients c 1, c 2, ..., c n the following equality holds:
In this form, Cramer's formula is valid without the assumption that Δ is different from zero; it is not even necessary that the coefficients of the system be elements of an integral ring (the determinant of the system can even be a zero divisor in the coefficient ring). We can also assume that either the sets b 1 ,b 2 ,...,b n And x 1 ,x 2 ,...,x n, or a set c 1 ,c 2 ,...,c n consist not of elements of the coefficient ring of the system, but of some module above this ring.
5.Minor of kth order. Matrix rank. Elementary transformations of matrices. The Kronecker-Capelli theorem on the compatibility conditions for a system of linear equations. Variable elimination (Gaussian) method for a system of linear equations.
Minor matrices A is the determinant of the square matrix of order k(which is also called the order of this minor), whose elements appear in the matrix A at the intersection of rows with numbers and columns with numbers.
Rank matrix row (column) system A With m lines and n columns is the maximum number of non-zero rows (columns).
Several rows (columns) are said to be linearly independent if none of them can be expressed linearly in terms of the others. The rank of the row system is always equal to the rank of the column system, and this number is called the rank of the matrix.
Kronecker - Capelli theorem (consistency criterion for a system of linear algebraic equations) -
a system of linear algebraic equations is consistent if and only if the rank of its main matrix is equal to the rank of its extended matrix (with free terms), and the system has a unique solution if the rank is equal to the number of unknowns, and an infinite number of solutions if the rank is less than the number of unknowns.
Gauss method - a classical method for solving a system of linear algebraic equations (SLAE). This is a method of sequential elimination of variables, when, using elementary transformations, a system of equations is reduced to an equivalent system of a step (or triangular) form, from which all other variables are found sequentially, starting with the last (by number) variables.
6. Directed segment and vector. Basic concepts of vector algebra. The sum of vectors and the product of a vector and a number. Condition for coordination of vectors. Properties of linear operations on vectors.
Operations on vectors
Addition
The operation of adding geometric vectors can be defined in different ways, depending on the situation and the type of vectors being considered:
Two vectors u, v and the vector of their sum
Triangle rule. To add two vectors and according to the triangle rule, both of these vectors are transferred parallel to themselves so that the beginning of one of them coincides with the end of the other. Then the sum vector is given by the third side of the resulting triangle, and its beginning coincides with the beginning of the first vector, and its end with the end of the second vector.
Parallelogram rule. To add two vectors and according to the parallelogram rule, both of these vectors are transferred parallel to themselves so that their origins coincide. Then the sum vector is given by the diagonal of the parallelogram constructed on them, starting from their common origin.
And the modulus (length) of the sum vector 
determined by the cosine theorem where is the angle between vectors when the beginning of one coincides with the end of the other. The formula is also used now - the angle between vectors emerging from one point.
Vector artwork
Vector artwork vector by vector is a vector that satisfies the following requirements:
Properties of vector C
§ the length of a vector is equal to the product of the lengths of the vectors and the sine of the angle φ between them
§ the vector is orthogonal to each of the vectors and
§ the direction of vector C is determined by the Buravchik rule
Properties of a vector product:
1. When rearranging the factors, the vector product changes sign (anticommutativity), i.e. 
2. The vector product has the combining property with respect to the scalar factor, that is
3. The vector product has the distribution property:
Basis and coordinate system on the plane and in space. Decomposition of a vector by basis. Orthonormal basis and rectangular Cartesian coordinate system on the plane and in space. Coordinates of a vector and a point on a plane and in space. Projections of a vector on the coordinate axes.
Basis (ancient Greek βασις, basis) - a set of vectors in a vector space such that any vector in this space can be uniquely represented as a linear combination of vectors from this set - basis vectors.
It is often convenient to choose the length (norm) of each of the basis vectors to be unit, such a basis is called normalized.
Representation of a specific (any) vector of space as a linear combination of basis vectors (the sum of basis vectors by numerical coefficients), for example
or, using the sum sign Σ:
called expansion of this vector over this basis.
Coordinates of a vector and a point on a plane and in space.
The x-axis coordinate of point A is a number equal in absolute value to the length of the segment OAx: positive if point A lies on the positive x-axis, and negative if it lies on the negative semi-axis.
A unit vector or unit vector is a vector whose length is equal to one and which is directed along any coordinate axis.
Then vector projection AB on the l axis is the difference x1 – x2 between the coordinates of the projections of the end and beginning of the vector onto this axis.
8.Length and direction cosines of a vector, relationship between direction cosines. Orth vector. Coordinates are the sum of vectors, the product of a vector and a number.
The vector length is determined by the formula
The direction of the vector is determined by the angles α, β, γ formed by it with the coordinate axes Ox, Oy, Oz. The cosines of these angles (the so-called direction cosines vector
) are calculated using the formulas: 
Unit vector or ort (unit vector of a normalized vector space) is a vector whose norm (length) is equal to one.
The unit vector, collinear with a given one (normalized vector), is determined by the formula
Unit vectors are often chosen as basis vectors, as this simplifies calculations. Such bases are called normalized. If these vectors are also orthogonal, such a basis is called an orthonormal basis.
Coordinates collinear
Coordinates equal
Coordinates sum vector two vectors satisfy the relations:
Coordinates collinear vectors satisfy the relation:
Coordinates equal vectors satisfy the relations:
Sum vector two vectors:
Sum of several vectors:
Product of a vector and a number:
Cross product of vectors. Geometric applications of cross product. Condition for collinearity of vectors. Algebraic properties of a mixed product. Expressing the vector product through the coordinates of the factors.
Cross product of a vector and vector b is called vector c, which:
1. Perpendicular to vectors a and b, i.e. c^a and c^b;
2. Has a length numerically equal to the area of a parallelogram constructed on vectors a and b as sides (see Fig. 17), i.e.
3.Vectors a, b and c form a right-handed triple.
Geometric Applications:
Establishing collinearity of vectors
Finding the area of a parallelogram and a triangle
According to the definition of the vector product of vectors A and b |a xb | =|a| * |b |sing, i.e. S pairs = |a x b |. And, therefore, DS =1/2|a x b |.
Determination of the moment of force about a point
It is known from physics that moment of force F relative to the point ABOUT called a vector M, which passes through the point ABOUT And:
1) perpendicular to the plane passing through the points O, A, B;
2) numerically equal to the product of force per arm
3) forms a right triple with vectors OA and A B.
Therefore, M = OA x F.
Finding linear rotation speed
The speed v of a point M of a rigid body rotating with an angular velocity w around a fixed axis is determined by the Euler formula v =w xr, where r =OM, where O is some fixed point of the axis (see Fig. 21).
Condition for collinearity of vectors - a necessary and sufficient condition for the collinearity of a non-zero vector and a vector is the existence of a number that satisfies the equality.
Algebraic properties of a mixed product
The mixed product of vectors does not change when the factors are rearranged circularly and changes sign to the opposite when two factors are interchanged, while maintaining its modulus.
The vector multiplication sign " " inside a mixed product can be placed between any of its factors.
A mixed product is distributive with respect to any of its factors: (for example) if , then
Expressing the cross product in terms of coordinates
right coordinate system
left coordinate system
12.Mixed product of vectors. The geometric meaning of a mixed product, the condition of coplanarity of vectors. Algebraic properties of a mixed product. Expressing a mixed product through the coordinates of the factors.
Mixed The product of an ordered triple of vectors (a,b,c) is the scalar product of the first vector and the vector product of the second vector and the third.
Algebraic properties of a vector product
Anticommutativity
Associativity with respect to multiplication by a scalar
Distributivity by addition
Jacobi identity. Runs in R3 and breaks in R7
The vector products of the basis vectors are found by definition
Conclusion
where are the coordinates of both the direction vector of the line and the coordinates of a point belonging to the line.
Normal vector of a line in a plane. The equation of a line passing through a given point perpendicular to a given vector. General equation of a straight line. Equations of a straight line with an angular coefficient. The relative position of two straight lines on a plane
Normal a vector of a line is any non-zero vector perpendicular to this line.
- equation of a line passing through a given point perpendicular to a given vector
Ax + Wu + C = 0- general equation of a line.
Line equation of the form y=kx+b
called equation of a straight line with slope, and the coefficient k is called the slope of this line.
Theorem. In the equation of a straight line with slope y=kx+b
the angular coefficient k is equal to the tangent of the angle of inclination of the straight line to the abscissa axis:
Mutual arrangement:
– general equations of two lines on the Oxy coordinate plane. Then
1) if , then the lines coincide;
2) if , then straight and parallel;
3) if , then the lines intersect.
Proof . The condition is equivalent to the collinearity of normal vectors of given lines:
Therefore, if , then the straight lines intersect.
If 
, then , , and the equation of the line takes the form:
Or 
, i.e. straight match. Note that the proportionality coefficient is , otherwise all coefficients of the general equation would be equal to zero, which is impossible.
If the lines do not coincide and do not intersect, then the case remains, i.e. straight parallel.
Equation of a line in segments
If in the general equation of the straight line Ах + Ву + С = 0 С≠0, then, dividing by –С, we get: or , where
The geometric meaning of the coefficients is that the coefficient A is the coordinate of the point of intersection of the line with the Ox axis, and b– the coordinate of the point of intersection of the straight line with the Oy axis.
Normal equation of a line
If both sides of the equation Ax + By + C = 0 are divided by a number called normalizing factor, then we get
xcosφ + ysinφ - p = 0 –
normal equation of a line.
The sign ± of the normalizing factor must be chosen so that μ ? WITH< 0.
p is the length of the perpendicular lowered from the origin to the straight line, and φ is the angle formed by this perpendicular with the positive direction of the Ox axis.
C It should be noted that not every line can be represented by an equation in segments, for example, lines parallel to axes or passing through the origin.
17. Ellipse. Canonical equation of an ellipse. Geometric properties and construction of an ellipse. Special terms.
Ellipse - locus of points M Euclidean plane, for which the sum of the distances to two given points F 1 and F 2 (called foci) is constant and greater than the distance between the foci, that is | F 1 M | + | F 2 M | = 2a, and | F 1 F 2 | < 2a.
Canonical equation
For any ellipse, you can find a Cartesian coordinate system such that the ellipse will be described by the equation (the canonical equation of the ellipse):
It describes an ellipse centered at the origin, whose axes coincide with the coordinate axes.
Construction: 1)Using a compass
2) Two tricks and a stretched thread
3) Ellipsograph (Ellipsograph consists of two sliders that can move along two perpendicular grooves or guides. The sliders are attached to the rod by means of hinges, and are located at a fixed distance from each other along the rod. The sliders move forward and backward - each along its own groove, - and the end of the rod describes an ellipse on the plane. The semi-axes of the ellipse a and b represent the distances from the end of the rod to the hinges on the sliders. Usually the distances a and b can be varied, and thereby change the shape and dimensions of the described ellipse)
Eccentricity characterizes the elongation of the ellipse. The closer the eccentricity is to zero, the more the ellipse resembles a circle, and vice versa, the closer the eccentricity is to unity, the more elongated it is.
Focal parameter
Canonical equation
18.Hyperbola. Canonical equations of hyperbolas. Geometric properties and construction of a hyperbola. Special terms
Hyperbola(ancient Greek ὑπερβολή, from ancient Greek βαλειν - “throw”, ὑπερ - “over”) - locus of points M Euclidean plane, for which the absolute value of the difference in distances from M up to two selected points F 1 and F 2 (called foci) constantly. More precisely,
Moreover | F 1 F 2 | > 2a > 0.
Ratios
For the characteristics of the hyperbolas defined above, they obey the following relations
2. The directrixes of the hyperbola are indicated by lines of double thickness and are indicated D 1 and D 2. Eccentricity ε equal to the ratio of the point distances P on the hyperbole to the focus and to the corresponding directrix (shown in green). The vertices of the hyperbola are designated as ± a. The hyperbola parameters mean the following:
a- distance from center C to each of the vertices
b- the length of the perpendicular dropped from each of the vertices to the asymptotes
c- distance from center C to any of the focuses, F 1 and F 2 ,
θ is the angle formed by each of the asymptotes and the axis drawn between the vertices.
Properties
§ For any point lying on a hyperbola, the ratio of the distances from this point to the focus to the distance from the same point to the directrix is a constant value.
§ A hyperbola has mirror symmetry about the real and imaginary axes, as well as rotational symmetry when rotated through an angle of 180° around the center of the hyperbola.
§ Each hyperbola has conjugate hyperbola, for which the real and imaginary axes change places, but the asymptotes remain the same. This corresponds to the replacement a And b on top of each other in a formula describing a hyperbola. The conjugate hyperbola is not the result of rotating the original hyperbola through an angle of 90°; both hyperbolas differ in shape.
19. Parabola. Canonical equation of a parabola. Geometric properties and construction of a parabola. Special terms.
Parabola - the geometric locus of points equidistant from a given line (called the directrix of a parabola) and a given point (called the focus of the parabola).
The canonical equation of a parabola in a rectangular coordinate system:
(or if you swap the axes).
Properties
§ 1 A parabola is a second order curve.
§ 2It has an axis of symmetry called parabola axis. The axis passes through the focus and is perpendicular to the directrix.
§ 3Optical property. A beam of rays parallel to the axis of the parabola, reflected in the parabola, is collected at its focus. And vice versa, light from a source located in focus is reflected by a parabola into a beam of rays parallel to its axis.
§ 4For a parabola, the focus is at the point (0.25; 0).
For a parabola, the focus is at the point (0; f).
§ 5 If the focus of a parabola is reflected relative to the tangent, then its image will lie on the directrix.
§ 6 A parabola is the antipoder of a line.
§ All parabolas are similar. The distance between the focus and the directrix determines the scale.
§ 7 When a parabola rotates around the axis of symmetry, an elliptical paraboloid is obtained.
Directrix of a parabola
Focal radius
20.Normal plane vector. The equation of a plane passing through a given point is perpendicular to a given vector. General plane equation, a special case of the general plane equation. Vector equation of a plane. The relative position of two planes.
Plane- one of the basic concepts of geometry. In a systematic presentation of geometry, the concept of plane is usually taken as one of the initial concepts, which is only indirectly determined by the axioms of geometry.
Equation of a plane by point and normal vector
In vector form
In coordinates
Angle between planes
Special cases of the general plane equation.
When studying various branches of physics, mechanics and technical sciences, quantities are encountered that are completely determined by specifying their numerical values. Such quantities are called scalar or, in short, scalars.
Scalar quantities are length, area, volume, mass, body temperature, etc. In addition to scalar quantities, in various problems there are quantities for which, in addition to their numerical value, it is also necessary to know their direction. Such quantities are called vector. Physical examples of vector quantities can be the displacement of a material point moving in space, the speed and acceleration of this point, as well as the force acting on it.
Vector quantities are represented using vectors.
Vector definition. A vector is a directed segment of a straight line that has a certain length.
A vector is characterized by two points. One point is the beginning point of the vector, the other point is the end point of the vector. If we denote the beginning of the vector with a dot A , and the end of the vector is a point IN , then the vector itself is denoted . A vector can also be denoted by one small Latin letter with a bar over it (for example, ).
Graphically, a vector is denoted by a segment with an arrow at the end.
The beginning of the vector is called its point of application. If the point A is the beginning of the vector , then we will say that the vector is applied at the point A.
A vector is characterized by two quantities: length and direction.
Vector length – the distance between the starting point A and the end point B. Another name for the length of a vector is the modulus of the vector and is indicated by the symbol . The vector modulus is denoted Vector , whose length is 1 is called a unit vector. That is, the condition for the unit vector
A vector with zero length is called a zero vector (denoted by ). Obviously, the zero vector has the same beginning and end points. The zero vector has no specific direction.
Definition of collinear vectors. Vectors and located on the same line or on parallel lines are called collinear .
Note that collinear vectors can have different lengths and different directions.
Determination of equal vectors. Two vectors are said to be equal if they are collinear, have the same length and the same direction.
In this case they write:
Comment. From the definition of equality of vectors it follows that a vector can be transferred in parallel by placing its origin at any point in space (in particular, a plane).
All zero vectors are considered equal.
Determination of opposite vectors. Two vectors are called opposite if they are collinear, have the same length, but the opposite direction.
In this case they write:
In other words, the vector opposite to the vector is denoted as .
To correctly display the laws of nature in physics, appropriate mathematical tools are required.
In geometry and physics there are quantities characterized by both numerical value and direction.
It is advisable to depict them as directed segments or vectors.
In contact with
Such quantities have a beginning (displayed by a dot) and an end, indicated by an arrow. The length of a segment is called (length).
- speed;
- acceleration;
- pulse;
- force;
- moment;
- strength;
- moving;
- field strength, etc.
Plane coordinates
Let us define a segment on the plane directed from point A (x1,y1) to point B (x2,y2). Its coordinates a (a1, a2) are the numbers a1=x2-x1, a2=y2-y1.
The module is calculated using the Pythagorean theorem: 
The beginning of the zero vector coincides with the end. The coordinates and length are 0.
Vector sum
Exist several rules for calculating the amount
- triangle rule;
- polygon rule;
- parallelogram rule.
The rule for adding vectors can be explained using problems from dynamics and mechanics. Let's consider the addition of vectors according to the triangle rule using the example of forces acting on a point body and successive movements of the body in space.
Let's say a body moves first from point A to point B, and then from point B to point C. The final displacement is a segment directed from the starting point A to the ending point C.
The result of two movements or their sum s = s1+ s2. This method is called triangle rule.
The arrows are lined up in a chain one after another, carrying out parallel transfer if necessary. The total segment closes the sequence. Its beginning coincides with the beginning of the first, its end with the end of the last. In foreign textbooks this method is called "tail to head".
The coordinates of the result c = a + b are equal to the sum of the corresponding coordinates of the terms c (a1+ b1, a2+ b2).
The sum of parallel (collinear) vectors is also determined by the triangle rule.
If two original segments are perpendicular to each other, then the result of their addition is the hypotenuse of the right triangle constructed on them. The length of the sum is calculated using the Pythagorean theorem.
Examples:
- The speed of a body thrown horizontally is perpendicular acceleration of free fall.
- With uniform rotational motion, the linear velocity of the body is perpendicular to the centripetal acceleration.
Addition of three or more vectors produce according to polygon rule, "tail to head"
Let us assume that forces F1 and F2 are applied to a point body.
Experience proves that the combined effect of these forces is equivalent to the action of one force directed along the diagonal of the parallelogram constructed on them. This resultant force is equal to their sum F = F1 + F 2. The above method of addition is called parallelogram rule.
The length in this case is calculated by the formula
Where θ is the angle between the sides.
The rules of triangle and parallelogram are interchangeable. In physics, the parallelogram rule is more often used, since the directional magnitudes of forces, velocities, and accelerations are usually applied to one point body. In a three-dimensional coordinate system, the parallelepiped rule applies.
Elements of algebra
- Addition is a binary operation: only a pair can be added at a time.
- Commutativity: the sum from the rearrangement of terms does not change a + b = b + a. This is clear from the parallelogram rule: the diagonal is always the same.
- Associativity: the sum of an arbitrary number of vectors does not depend on the order of their addition (a + b) + c = a + (b + c).
- Summation with a zero vector does not change either direction or length: a +0= a .
- For each vector there is opposite. Their sum is equal to zero a +(-a)=0, and the lengths are the same.
Multiplication by a scalar
The result of multiplication by a scalar is a vector.
The coordinates of the product are obtained by multiplying by a scalar the corresponding coordinates of the original.
A scalar is a numerical value with a plus or minus sign, greater or less than one.
Examples of scalar quantities in physics:
- weight;
- time;
- charge;
- length;
- square;
- volume;
- density;
- temperature;
- energy.
Example:
Work is the scalar product of force and displacement A = Fs.