Vector projection and its notation. Projection (geometric, algebraic) of a vector onto an axis

§ 3. Projections of a vector on the coordinate axes

1. Finding projections geometrically.

Vector
- projection of the vector onto the axis OX
- projection of the vector onto the axis OY

Definition 1. Vector projection on any coordinate axis is a number taken with a plus or minus sign, corresponding to the length of the segment located between the bases of the perpendiculars dropped from the beginning and end of the vector to the coordinate axis.

The projection sign is defined as follows. If, when moving along the coordinate axis, there is a movement from the projection point of the beginning of the vector to the projection point of the end of the vector in the positive direction of the axis, then the projection of the vector is considered positive. If it is opposite to the axis, then the projection is considered negative.

The figure shows that if the vector is oriented somehow opposite to the coordinate axis, then its projection onto this axis is negative. If a vector is oriented somehow in the positive direction of the coordinate axis, then its projection onto this axis is positive.

If a vector is perpendicular to the coordinate axis, then its projection onto this axis is zero.
If a vector is codirectional with an axis, then its projection onto this axis is equal to the absolute value of the vector.
If a vector is directed oppositely to the coordinate axis, then its projection onto this axis is equal in absolute value to the absolute value of the vector taken with a minus sign.

2. Most general definition projections.

From a right triangle ABD: .

Definition 2. Vector projection on any coordinate axis is a number equal to the product of the modulus of the vector and the cosine of the angle formed by the vector with the positive direction of the coordinate axis.

The sign of the projection is determined by the sign of the cosine of the angle formed by the vector with the positive axis direction.
If the angle is acute, then the cosine has a positive sign and the projections are positive. For obtuse angles, the cosine has a negative sign, so in such cases the projections onto the axis are negative.
- therefore, for vectors perpendicular to the axis, the projection is zero.

Projection 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 – scalar projection vector A to the X axis, then a x i- its vector projection onto this axis.

Let's denote vector projection the same as the vector itself, but with the index of the axis on which the vector is projected. So, the vector projection of the vector A on the X axis we denote A x ( fat a letter denoting a vector and a subscript of the axis name) or (a 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, absolute value which is equal to the length of the axis segment (on the selected scale) enclosed between the projections of the start point and the end point of the vector. Usually instead of the expression scalar projection they simply say - projection. The projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a lower index (as a rule) of the name of the axis on which this vector is projected. For example, if a 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 axis is Y, its projection will be denoted a y.

To calculate the projection vector on an axis (for example, the X axis), it is necessary to subtract the coordinate of the starting point from the coordinate of its end point, that is
a x = x k − x n.
The projection of a vector onto 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 equals 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.

From the figure it is clear 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 and the cosine of the angle between the direction of the axis and vector direction. If the angle is acute, then
Cos α > 0 and a x > 0, and, if obtuse, then the cosine of the obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

Angles measured from the axis counterclockwise are considered positive, and angles measured along the axis are negative. However, since cosine is an even function, that is, Cos α = Cos (− α), when calculating projections, 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.

Vector coordinates— coefficients of the only possible linear combination basis vectors in the selected coordinate system equal to the given vector.

where are the coordinates of the vector.

Dot product of vectors

Scalar product of vectors[- in finite-dimensional vector space is defined as the sum of the products of identical components being multiplied vectors.

For example, S.p.v. a = (a 1 , ..., a n) And b = (b 1 , ..., b n):

(a , b ) = a 1 b 1 + a 2 b 2 + ... + a n b n

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

1. Value of vector and scalar…………………………………….4

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

3. Projection of the vector onto the axis………………………………………………………...6

4. Basic formula vector algebra……………………………..8

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

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

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

Introduction:

Physics is inextricably linked with mathematics. Mathematics gives physics the means and techniques for a general and precise expression of the relationship between physical quantities, which are discovered as a result of experiment or theoretical research. After all, the main method of research in physics is experimental. This means that a scientist reveals calculations using measurements. Denotes the relationship between various physical quantities. Then, everything is translated into the language of mathematics. Formed mathematical model. Physics is a science that studies the simplest and at the same time the most general laws. The task of physics is to create such a picture in our minds physical world, which most fully reflects its properties and provides such relationships between the elements of the model that 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 a particular 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 - to choose the main thing from all the diversity.

Physical models are mathematical, but mathematics is not their basis. Quantitative relationships between physical quantities are determined 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. Meaning of 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, electric and magnetic field strength. They can be contrasted with other quantities such as mass, volume, pressure, temperature and density, which can be described by an ordinary number, and are called " scalars".

They are written either in regular font letters or in numbers (a, b, t, G, 5, −7....). Scalar quantities can be positive or negative. At the same time, some objects of study may have such properties that full description For which knowledge of only a numerical measure turns out to be insufficient, it is also 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 by a letter in regular (non-bold) font, but with an arrow above it:

In addition, a vector is often denoted by a pair of letters (usually capitalized), with the first letter indicating the beginning of the vector and the second 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 in exactly the same way as a vector (that is, in bold or regular, but with arrow), but then the vector designation 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 true a= b. It should also be borne in mind that the vector symbol may be preceded by a minus sign, for example - c, however, this sign symbolically indicates that the vector -c has the same modulus as the vector c, but is directed towards the opposite side.

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

2. Determination of the projection, axis and coordinate of the point.

Axis- This is a straight line that is given some direction.
An axis is designated by some letter: X, Y, Z, s, t... Usually a point is selected (arbitrarily) on the axis, which is called the origin and, as a rule, is designated by the letter O. From this point the distances to other points of interest to us are measured.

Projection of a point on an axis is the base of a perpendicular drawn from this point onto a given axis. That is, the projection of a point onto the axis is a point.

Point coordinate on a given axis is a number whose absolute value is equal to the length of the axis segment (on the selected scale) contained between the origin of the axis and the projection of the point onto this axis. This number is taken with a plus sign if the projection of the 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 vector a onto the X axis, then a x ·i is its vector projection onto this axis.

Let us denote the vector projection in the same way as the vector itself, but with the index of the axis on which the vector is projected. Thus, we denote the vector projection of vector a onto the X axis as a x (a bold letter denoting the vector and the subscript of the axis name) or

(a low-bold letter denoting a vector, but with an arrow at the top (!) and a subscript for 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 (on the selected scale) enclosed between the projections of the start point and the end point of the vector. Usually instead of the expression scalar projection they simply say - projection. The projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a lower index (as a rule) of the name of the axis on which this vector is projected. For example, if a 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 axis is Y, its projection will be denoted a y.

To calculate the projection vector on an axis (for example, the X axis), it is necessary to subtract the coordinate of the starting point from the coordinate of its end point, that is

a x = x k − x n.

The projection of a vector onto 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 equals 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.

From the figure it is clear 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 and the cosine of the angle between the direction of the axis and vector direction. If the angle is acute, then
Cos α > 0 and a x > 0, and, if obtuse, then the cosine of the obtuse angle is negative, and the projection of the vector onto the axis will also be negative.

Angles measured from the axis counterclockwise are considered positive, and angles measured along the axis are negative. However, since cosine is an even function, that is, Cos α = Cos (− α), when calculating projections, 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. Basic formula of vector algebra.

Let's project vector a on the X and Y axes of the rectangular coordinate system. Let's find the vector projections of vector a on these axes:

a x = a x ·i, and y = a y ·j.

But in accordance with the rule of vector addition

a = a x + a y.

a = a x i + a y j.

Thus, we have expressed a vector in terms of its projections and vectors of the 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 we performed is called the decomposition of a 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 formula of vector algebra. Of course, it can be written like this.

Answer:

Projection properties:

Vector Projection Properties

Property 1.

The projection of the sum of two vectors onto an axis is equal to the sum of the projections of vectors onto the same axis:

This property allows you to replace the projection of a sum of vectors with the sum of their projections and vice versa.

Property 2. If a vector is multiplied by the number λ, then its projection onto the axis is also multiplied by this number:

Property 3.

The projection of the vector onto the l axis is equal to the product of the modulus of the vector and the cosine of the angle between the vector and the axis:

Orth axis. Decomposition of a vector in coordinate unit vectors. Vector coordinates. Coordinate properties

Answer:

Unit vectors of the axes.

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

In the three-dimensional case, the unit vectors are usually denoted

And Arrow symbols and may also be used.

In this case, in the case of a right coordinate system, the following formulas with vector products of unit vectors are valid:

Decomposition of a vector in coordinate unit vectors.

The unit vector of the coordinate axis is denoted by , axes by , 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 coordinate axes has the form:

Vector coordinates:

To calculate the coordinates of a vector, knowing the coordinates (x1; y1) of its beginning A and the coordinates (x2; y2) of its end B, you need to subtract the coordinates of the beginning from the coordinates of the end: (x2 – x1; y2 – y1).

Properties of coordinates.

Consider a coordinate line with the origin at point O and the unit vector i. Then for any vector a on this line: a = axi.

The number ax is called the coordinate of the vector a on the coordinate axis.

Property 1. When adding vectors on an axis, their coordinates are added.

Property 2. When a vector is multiplied by a number, its coordinate is multiplied by that number.

Dot product of vectors. Properties.

Answer:

The scalar product of two non-zero vectors is the number

equal to the product of these vectors and the cosine of the angle between them.

Properties:

1. The scalar product has the commutative property: ab=ba

Scalar product of coordinate unit vectors. Determination of the scalar product of vectors specified by their coordinates.

Answer:

Dot product (×) of unit vectors

(X)	I	J	K
I
J
K

Determination of the scalar product of vectors specified by their coordinates.

The scalar product of two vectors and given by their coordinates can be calculated using the formula

The cross product of two vectors. Properties of a vector product.

Answer:

Three non-coplanar vectors form a right triple if from the end of the third the rotation from the first vector to the second is made counterclockwise. If clockwise, then left. If not, then in the opposite direction ( show how he showed with “handles”)

Cross product of a vector A to vector b called a vector from which:

1. Perpendicular to vectors A And b

2. Has length, numerically equal to the area parallelogram formed on a And b vectors

3. Vectors, a ,b, And c form a right-hand triplet of vectors

Properties:

1.

3.

4.

Vector product of coordinate unit vectors. Determination of the vector product of vectors specified by their coordinates.

Answer:

Vector product of coordinate unit vectors.

Determination of the vector product of vectors specified by their coordinates.

Let the vectors a = (x1; y1; z1) and b = (x2; y2; z2) be given by their coordinates in a rectangular Cartesian system coordinates O, i, j, k, and the triple i, j, k is right.

Let's expand a and b into basis vectors:

a = x 1 i + y 1 j + z 1 k, b = x 2 i + y 2 j + z 2 k.

Using the properties of the vector product, we get

[A; b] = =

= x 1 x 2 + x 1 y 2 + x 1 z 2 +

+ y 1 x 2 + y 1 y 2 + y 1 z 2 +

+ z 1 x 2 + z 1 y 2 + z 1 z 2 . (1)

By the definition of a vector product we find

= 0, = k, = - j,

= - k, = 0, = i,

= j, = - i. = 0.

Taking these equalities into account, formula (1) can be written as follows:

[A; b] = x 1 y 2 k - x 1 z 2 j - y 1 x 2 k + y 1 z 2 i + z 1 x 2 j - z 1 y 2 i

[A; b] = (y 1 z 2 - z 1 y 2) i + (z 1 x 2 - x 1 z 2) j + (x 1 y 2 - y 1 x 2) k. (2)

Formula (2) gives an expression for the vector product of two vectors specified by their coordinates.

The resulting formula is cumbersome. Using the notation of determinants, you can write it in another form that is more convenient for memorization:

Usually formula (3) is written even shorter:

Definition 1. On a plane, a parallel projection of point A onto the l axis is a point - the point of intersection of the l axis with a straight line drawn through point A parallel to the vector that specifies the design direction.

Definition 2. The parallel projection of a vector onto the l axis (to the vector) is the coordinate of the vector relative to the basis axis l, where points and are parallel projections of points A and B onto the l axis, respectively (Fig. 1).

According to the definition we have

Definition 3. if and l axis basis Cartesian, that is, the projection of the vector onto the l axis called orthogonal (Fig. 2).

In space, definition 2 of the vector projection onto the axis remains in force, only the projection direction is specified by two non-collinear vectors (Fig. 3).

From the definition of the projection of a vector onto an axis it follows that each coordinate of a vector is a projection of this vector onto the axis defined by the corresponding basis vector. In this case, the design direction is specified by two other basis vectors if the design is carried out (considered) in space, or by another basis vector if the design is considered on a plane (Fig. 4).

Theorem 1. The orthogonal projection of a vector onto the l axis is equal to the product of the modulus of the vector and the cosine of the angle between the positive direction of the l axis and, i.e.

On the other side

From we find

Substituting AC into equality (2), we obtain

Since the numbers x and the same sign in both cases under consideration ((Fig. 5, a) ; (Fig. 5, b), then from equality (4) it follows

Comment. In what follows, we will consider only the orthogonal projection of the vector onto the axis and therefore the word “ort” (orthogonal) will be omitted from the notation.

Let us present a number of formulas that are used later in solving problems.

a) Projection of the vector onto the axis.

If, then the orthogonal projection onto the vector according to formula (5) has the form

c) Distance from a point to a plane.

Let b be a given plane with a normal vector, M be a given point,

d is the distance from point M to plane b (Fig. 6).

If N- arbitrary point planes b, a and are projections of points Mi and N onto the axis, then

• G) The distance between intersecting lines.

Let a and b be given crossing lines, be a vector perpendicular to them, A and B be arbitrary points of lines a and b, respectively (Fig. 7), and and be projections of points A and B onto, then

e) Distance from a point to a line.

Let l- a given straight line with a direction vector, M - a given point,

N - its projection onto the line l, then - the required distance (Fig. 8).

If A is an arbitrary point on a line l, then in right triangle MNA, hypotenuse MA and legs can be found. Means,

f) The angle between a straight line and a plane.

Let be the direction vector of this line l, - normal vector of a given plane b, - projection of a straight line l to plane b (Fig. 9).

As is known, the angle μ between a straight line l and its projection onto plane b is called the angle between the line and the plane. We have

Let us give examples of solving metric problems using the vector-coordinate method.