Find vector projection online calculator. Online calculator. Calculating the projection of a vector on a vector

and on an axis or some other vector there are concepts of its geometric projection and numerical (or algebraic) projection. The result of a geometric projection is a vector, and the result of an algebraic projection is a non-negative real number. But before moving on to these concepts, let's remember the necessary information.

Preliminary information

The main concept is the concept of a vector itself. In order to introduce the definition of a geometric vector, let's recall what a segment is. Let us introduce the following definition.

Definition 1

A segment is a part of a straight line that has two boundaries in the form of points.

A segment can have 2 directions. To indicate the direction, we will call one of the boundaries of the segment its beginning, and the other boundary - its end. The direction is indicated from its beginning to the end of the segment.

Definition 2

A vector or a directed segment is a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.

Designation: Two letters: $ \ overline (AB) $ - (where $ A $ is its beginning and $ B $ is its end).

One small letter: $ \ overline (a) $ (fig. 1).

Let us introduce a few more concepts related to the concept of a vector.

Definition 3

Two nonzero vectors will be called collinear if they lie on the same straight line or on straight lines parallel to each other (Fig. 2).

Definition 4

Two nonzero vectors will be called codirectional if they satisfy two conditions:

These vectors are collinear.
If they point in one direction (fig. 3).

Designation: $ \ overline (a) \ overline (b) $

Definition 5

Two nonzero vectors will be called oppositely directed if they satisfy two conditions:

These vectors are collinear.
If they are directed in different directions (fig. 4).

Designation: $ \ overline (a) ↓ \ overline (d) $

Definition 6

The length of the vector $ \ overline (a) $ is the length of the segment $ a $.

Notation: $ | \ overline (a) | $

Let us turn to the definition of the equality of two vectors

Definition 7

Two vectors will be called equal if they satisfy two conditions:

They are co-directed;
Their lengths are equal (Fig. 5).

Geometric projection

As we said earlier, the result of a geometric projection will be a vector.

Definition 8

The geometric projection of the vector $ \ overline (AB) $ onto an axis is a vector that is obtained as follows: The origin of the vector $ A $ is projected onto this axis. We get the point $ A "$ - the beginning of the desired vector. The end point of the vector $ B $ is projected onto this axis. We get the point $ B" $ - the end of the desired vector. The vector $ \ overline (A "B") $ will be the desired vector.

Consider the problem:

Example 1

Construct a geometric projection $ \ overline (AB) $ on the $ l $ axis, shown in Figure 6.

Draw from point $ A $ a perpendicular to the axis $ l $, we get the point $ A "$ on it. Next, draw from point $ B $ a perpendicular to the axis $ l $, we get point $ B" $ on it (Fig. 7).

Introduction ………………………………………………………………………… 3

1. Vector and scalar value ………………………………………… .4

2. Determination of projection, axis and coordinate of a point ……………… ... 5

3. Vector projection on the axis ………………………………………… ... 6

4. The basic formula of vector algebra …………………………… ..8

5. Calculation of the modulus of a vector by its projections ………………… ... 9

Conclusion ………………………………………………………………… ... 11

Literature ………………………………………………………………… ... 12

Introduction:

Physics is inextricably linked with mathematics. Mathematics provides physics with the means and techniques for a general and accurate expression of the relationship between physical quantities, which are discovered as a result of experiment or theoretical research, because the main research method in physics is experimental. This means that the scientist identifies calculations using measurements. Indicates the relationship between various physical quantities. Then, everything is translated into the language of mathematics. A mathematical model is being formed. Physics is a science that studies the simplest and at the same time the most general laws. The task of physics is to create in our consciousness such a picture of the physical world that most fully reflects its properties and provides such relationships between the elements of the model as exist between the elements.

So, physics creates a model of the world around us and studies its properties. But any model is limited. When creating models of this or that phenomenon, only properties and connections that are essential for a given range of phenomena are taken into account. This is the art of a scientist - from all the diversity to choose the main thing.

Physical models are mathematical, but mathematics is not their basis. Quantitative relationships between physical quantities are found out as a result of measurements, observations and experimental research and are only expressed in the language of mathematics. However, there is no other language for constructing physical theories.

1. The value of the vector and scalar.

In physics and mathematics, a vector is a quantity that is characterized by its numerical value and direction. In physics, there are many important quantities that are vectors, for example, force, position, speed, acceleration, torque, momentum, strength of electric and magnetic fields. They can be contrasted with other quantities such as mass, volume, pressure, temperature and density, which can be described by the usual number, and they are called " scalars " .

They are written either in letters of a regular font, or in numbers (a, b, t, G, 5, −7 ....). Scalars can be positive or negative. At the same time, some objects of study may have such properties for full description whose knowledge of only a numerical measure turns out to be insufficient, it is still necessary to characterize these properties by direction in space. Such properties are characterized by vector quantities (vectors). Vectors, unlike scalars, are denoted by bold letters: a, b, g, F, C….
Often, a vector is denoted with a letter in a regular (non-bold) font, but with an arrow above it:

In addition, a vector is often denoted by a pair of letters (usually capital letters), with the first letter denoting the beginning of the vector, and the second denoting its end.

The modulus of a vector, that is, the length of a directed straight line segment, is denoted by the same letters as the vector itself, but in normal (not bold) writing and without an arrow above them, or just like the vector (that is, in bold or normal, but with arrow), but then the designation of the vector is enclosed in vertical dashes.
A vector is a complex object that is simultaneously characterized by both magnitude and direction.

There are also no positive and negative vectors. But vectors can be equal to each other. This is when, for example, a and b have the same modules and are directed in the same direction. In this case, the notation is valid a= b. It should also be borne in mind that a minus sign can appear in front of the vector symbol, for example, - c, however, this sign symbolically indicates that the vector -c has the same modulus as the vector c, but is directed in the opposite direction.

The vector -c is called the opposite (or inverse) of the vector c.
In physics, however, each vector is filled with specific content, and when comparing vectors of the same type (for example, forces), the points of their application can be of significant importance.

2. Determination of projection, axis and point coordinate.

Axis- this is a straight line, which is given some direction.
The axis is denoted by any letter: X, Y, Z, s, t ... Usually on the axis a point is chosen (arbitrarily), which is called the origin and, as a rule, is denoted by the letter O. Distances to other points of interest to us are counted from this point.

Point projection on an axis is called the base of the perpendicular dropped from this point on this axis. That is, the projection of a point onto an axis is a point.

Coordinate point on a given axis, a number is called, the absolute value of which is equal to the length of the axis segment (in the selected scale), enclosed between the origin of the axis and the projection of a point on this axis. This number is taken with a plus sign if the projection of a point is located in the direction of the axis from its origin and with a minus sign if in the opposite direction.

3.Projection of the vector onto the axis.

The projection of a vector onto an axis is a vector that is obtained by multiplying the scalar projection of a vector onto this axis and the unit vector of this axis. For example, if a x is the scalar projection of the vector a onto the X axis, then a x i is its vector projection onto this axis.

We denote the vector projection in the same way as the vector itself, but with the index of the axis onto which the vector is projected. So, the vector projection of the vector a onto the X axis is denoted by a x (bold letter denoting the vector and the subscript of the axis name) or

(non-bold letter denoting a vector, but with an arrow at the top (!) and a subscript of the axis name).

Scalar projection vector per axis is called number, the absolute value of which is equal to the length of the axis segment (in the selected scale), enclosed between the projections of the start point and the end point of the vector. Usually, instead of expressing scalar projection they just say - projection... The projection is denoted by the same letter as the projected vector (in normal, non-bold notation), with a subscript (usually) of the name of the axis onto which this vector is projected. For example, if the vector is projected onto the X-axis a, then its projection is denoted by a x. When projecting the same vector onto another axis, if the Y axis, its projection will be denoted by a y.

To compute the projection vector on the axis (for example, the X axis), subtract the coordinate of the start point from the coordinate of the point of its end, that is

a x = x k - x n.

The projection of a vector to an axis is a number. Moreover, the projection can be positive if the value x k is greater than the value x n,

negative if the value x k is less than the value x n

and equal to zero if x k is equal to x n.

The projection of a vector onto an axis can also be found by knowing the modulus of the vector and the angle it makes with this axis.

The figure shows that a x = a Cos α

That is, the projection of the vector onto the axis is equal to the product of the modulus of the vector by the cosine of the angle between the direction of the axis and vector direction... If the angle is sharp, then
Cos α> 0 and a x> 0, and if it is obtuse, then the cosine of the obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

Angles counted from the axis counterclockwise are considered to be positive, and along the way - negative. However, since the cosine is an even function, that is, Cos α = Cos (- α), then when calculating the projections, the angles can be counted both clockwise and counterclockwise.

To find the projection of a vector onto an axis, the modulus of this vector must be multiplied by the cosine of the angle between the direction of the axis and the direction of the vector.

4. The main formula of vector algebra.

Project vector a on the X and Y axes of a rectangular coordinate system. Let us find the vector projections of the vector a on these axes:

a x = a x i, and y = a y j.

But in accordance with the vector addition rule

a = a x + a y.

a = a x i + a y j.

Thus, we have expressed the vector in terms of its projections and unit vectors of a rectangular coordinate system (or in terms of its vector projections).

Vector projections a x and a y are called components or components of the vector a. The operation that we have performed is called the expansion of the vector along the axes of a rectangular coordinate system.

If the vector is given in space, then

a = a x i + a y j + a z k.

This formula is called the basic vector algebra formula. Of course, it can be written like this.

Let two vectors and be given in space. Set aside from an arbitrary point O vectors and. Corner between vectors and is called the smallest of the angles. Denoted .

Consider the axis l and put a unit vector on it (i.e., a vector whose length is equal to one).

Angle between vector and axis l understand the angle between vectors and.

So let l- some axis and - vector.

Let us denote by A 1 and B 1 axis projection l respectively points A and B... Let's pretend that A 1 has a coordinate x 1, a B 1- coordinate x 2 on the axis l.

Then projection vectors per axis l called the difference x 1 – x 2 between the coordinates of the projections of the end and the beginning of the vector on this axis.

Projection of a vector onto an axis l will denote.

It is clear that if the angle between the vector and the axis l sharp then x 2> x 1, and the projection x 2 – x 1> 0; if this angle is obtuse, then x 2< x 1 and projection x 2 – x 1< 0. Наконец, если вектор перпендикулярен оси l, then x 2= x 1 and x 2– x 1=0.

Thus, the projection of the vector onto the axis l Is the length of the segment A 1 B 1 taken from a certain sign... Therefore, the projection of a vector onto an axis is a number or a scalar.

The projection of one vector onto another is determined in a similar way. In this case, the projections of the ends of the given vector on the line on which the 2nd vector lies are found.

Let's look at some of the main projection properties.

LINEAR DEPENDENT AND LINEAR INDEPENDENT VECTOR SYSTEMS

Let's consider several vectors.

Linear combination of these vectors is called any vector of the form, where are some numbers. The numbers are called the coefficients of the linear combination. They also say that in this case it is linearly expressed in terms of these vectors, i.e. is obtained from them using linear actions.

For example, if three vectors are given, then vectors can be considered as their linear combination:

If a vector is presented as a linear combination of some vectors, then they say that it decomposed along these vectors.

The vectors are called linearly dependent if there are numbers, not all equal to zero, such that ... It is clear that the given vectors will be linearly dependent if any of these vectors is linearly expressed in terms of the others.

Otherwise, i.e. when the ratio is performed only when , these vectors are called linearly independent.

Theorem 1. Any two vectors are linearly dependent if and only if they are collinear.

Proof:

The following theorem can be proved similarly.

Theorem 2. Three vectors are linearly dependent if and only if they are coplanar.

Proof.

BASIS

Basis is called a set of nonzero linearly independent vectors. The elements of the basis will be denoted by.

In the previous section, we saw that two non-collinear vectors in the plane are linearly independent. Therefore, according to Theorem 1, from the previous section, a basis on a plane is any two non-collinear vectors on this plane.

Similarly, any three non-coplanar vectors are linearly independent in space. Consequently, three non-coplanar vectors are called a basis in space.

The following statement is true.

Theorem. Let a basis be given in space. Then any vector can be represented as a linear combination , where x, y, z- some numbers. This decomposition is unique.

Proof.

Thus, the basis makes it possible to unambiguously associate each vector with a triplet of numbers - the coefficients of the expansion of this vector in terms of the vectors of the basis:. The converse is also true, for every triplet of numbers x, y, z using a basis, you can match a vector if you compose a linear combination .

If the basis and , then the numbers x, y, z are called coordinates vectors in the given basis. Vector coordinates denote.

DECART'S SYSTEM OF COORDINATES

Let a point be given in space O and three non-coplanar vectors.

Cartesian coordinate system in space (on a plane) is called a set of a point and a basis, i.e. a set of a point and three non-coplanar vectors (2 non-collinear vectors) outgoing from this point.

Point O called the origin; straight lines passing through the origin in the direction of the basis vectors are called coordinate axes - the abscissa, ordinate and applicate axes. The planes passing through the coordinate axes are called coordinate planes.

Consider an arbitrary point in the selected coordinate system M... Let's introduce the concept of point coordinates M... The vector connecting the origin to the point M... called radius vector points M.

A vector in the selected basis can be associated with a triple of numbers - its coordinates: .

Point radius vector coordinates M... are called coordinates of point M... in the considered coordinate system. M (x, y, z)... The first coordinate is called the abscissa, the second is the ordinate, and the third is the applicate.

Cartesian coordinates on the plane are determined in a similar way. Here the point has only two coordinates - the abscissa and the ordinate.

It is easy to see that for a given coordinate system, each point has certain coordinates. On the other hand, for every triplet of numbers, there is a single point that has these numbers as coordinates.

If the vectors taken as a basis in the selected coordinate system have unit length and are pairwise perpendicular, then the coordinate system is called Cartesian rectangular.

It is easy to show that.

Direction cosines of a vector completely define its direction, but say nothing about its length.

Algebraic vector projection on any axis is equal to the product of the length of the vector by the cosine of the angle between the axis and the vector:

Pr a b = | b | cos (a, b) or

Where a b is the scalar product of vectors, | a | is the modulus of the vector a.

Instruction. To find the projection of the vector Pp a b into online mode you must specify the coordinates of vectors a and b. In this case, the vector can be specified on a plane (two coordinates) and in space (three coordinates). The resulting solution is saved in a Word file. If vectors are specified through the coordinates of points, then this calculator must be used.

Vector projection classification

Types of projections by definition vector projection

Coordinate projection views

Vector projection properties

The geometric projection of a vector is a vector (has a direction).
The algebraic projection of a vector is a number.

Vector projection theorems

Theorem 1. The projection of the sum of vectors onto any axis is equal to the projection of the terms of the vectors onto the same axis.

Theorem 2. The algebraic projection of a vector onto any axis is equal to the product of the length of the vector and the cosine of the angle between the axis and the vector:

Pr a b = | b | cos (a, b)

Types of vector projections

projection on the OX axis.
projection onto the OY axis.
vector projection.

OX projection	OY-axis projection	Vector projection
If the direction of the vector A'B 'coincides with the direction of the OX axis, then the projection of the vector A'B' has a positive sign.	If the direction of the vector A'B 'coincides with the direction of the OY axis, then the projection of the vector A'B' has a positive sign.	If the direction of the vector A'B 'coincides with the direction of the vector NM, then the projection of the vector A'B' has a positive sign.
If the direction of the vector is opposite to the direction of the OX axis, then the projection of the vector A'B 'has negative sign.	If the direction of the vector A'B 'is opposite to the direction of the OY axis, then the projection of the vector A'B' has a negative sign.	If the direction of vector A'B 'is opposite to the direction of vector NM, then the projection of vector A'B' has a negative sign.
If the vector AB is parallel to the OX axis, then the projection of the vector A'B 'is equal to the absolute value of the vector AB.	If the vector AB is parallel to the OY axis, then the projection of the vector A'B 'is equal to the absolute value of the vector AB.	If vector AB is parallel to vector NM, then the projection of vector A'B 'is equal to the modulus of vector AB.
If the vector AB is perpendicular to the OX axis, then the projection A'B 'is equal to zero (zero-vector).	If the vector AB is perpendicular to the OY axis, then the projection A'B 'is equal to zero (zero-vector).	If vector AB is perpendicular to vector NM, then projection A'B 'is equal to zero (zero-vector).

1. Question: Can the vector projection have a negative sign. Answer: Yes, the vector projection can be negative. In this case, the vector has the opposite direction (see how the OX axis and the AB vector are directed)
2. Question: Can the projection of the vector be the same as the modulus of the vector. Answer: Yes, it can. In this case, the vectors are parallel (or collinear).
3. Question: Can the projection of a vector be equal to zero (zero-vector). Answer: Yes, it can. In this case, the vector is perpendicular to the corresponding axis (vector).

Example 1. The vector (Fig. 1) forms an angle of 60 ° with the OX axis (it is specified by the vector a). If OE is a scale unit, then | b | = 4, so .

Indeed, the length of the vector (geometric projection b) is 2, and the direction coincides with the direction of the OX axis.

Example 2. The vector (Fig. 2) forms an angle (a, b) = 120 o with the OX axis (with the vector a). Length | b | vector b is equal to 4, therefore pr a b = 4 · cos120 o = -2.

Indeed, the length of the vector is 2, and the direction is opposite to the direction of the axis.

Designing various lines and surfaces on a plane allows you to build a visual image of objects in the form of a drawing. We will consider rectangular design, in which the projection rays are perpendicular to the projection plane. THE PROJECTION OF THE VECTOR ON THE PLANE consider the vector = (Fig. 3.22), enclosed between the perpendiculars omitted from its beginning and end.

Rice. 3.22. Vector projection of a vector onto a plane.

Rice. 3.23. Vector projection of the vector onto the axis.

In vector algebra, it is often necessary to project a vector onto the AXIS, that is, onto a straight line with a certain orientation. This design is easy if the vector and the L-axis lie in the same plane (Figure 3.23). However, the task becomes more difficult when this condition is not met. Let's construct the projection of the vector onto the axis when the vector and the axis do not lie in the same plane (Fig. 3.24).

Rice. 3.24. Projecting a vector to an axis
in general.

Through the ends of the vector we draw planes perpendicular to the straight line L. At the intersection with this straight line, these planes define two points A1 and B1 - a vector, which we will call the vector projection of this vector. The problem of finding the vector projection can be solved easier if the vector is brought into the same plane with the axis, which can be done, since free vectors are considered in vector algebra.

Along with the vector projection, there is also the SCALAR PROJECTION, which is equal to the modulus of the vector projection if the vector projection coincides with the orientation of the L axis, and is equal to the opposite value if the vector projection and the L axis have opposite orientations. The scalar projection will be denoted by:

Vector and scalar projections are not always terminologically strictly separated in practice. Usually the term "vector projection" is used, meaning the scalar vector projection. When deciding, it is necessary to clearly distinguish between these concepts. Following the established tradition, we will use the terms "vector projection", meaning scalar projection, and "vector projection" - in accordance with the established meaning.

Let us prove a theorem that allows calculating the scalar projection of a given vector.

THEOREM 5. The projection of a vector onto the L axis is equal to the product of its modulus by the cosine of the angle between the vector and the axis, that is

(3.5)

Rice. 3.25. Finding vector and scalar
Vector projections on the L axis
(and the L axis is equally oriented).

PROOF. Let's pre-construct to find the angle G Between the vector and the L-axis. To do this, construct a straight line MN, parallel to the L-axis and passing through the point O - the beginning of the vector (Fig. 3.25). The angle will be the desired angle. Let's draw through points A and O two planes perpendicular to the axis L. We get:

Since the L-axis and line MN are parallel.

We highlight two cases mutual disposition vector and L-axis.

1. Let the vector projection and the L-axis be identically oriented (Fig. 3.25). Then the corresponding scalar projection .

2. Let and L are oriented in different directions (fig. 3.26).

Rice. 3.26. Finding the vector and scalar projections of a vector onto the L-axis (and the L-axis are oriented in opposite directions).

Thus, the assertion of the theorem is true in both cases.

THEOREM 6. If the origin of the vector is reduced to some point on the L axis, and this axis is located in the s plane, the vector forms an angle with the vector projection onto the s plane, and an angle with the vector projection onto the L axis; in addition, the vector projections themselves form an angle between themselves , then