§ 3. Vector projections 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 called a number taken with a "plus" or "minus" sign, corresponding to the length of the segment located between the bases of the perpendiculars, lowered 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 - is opposite to the axis, then the projection is considered negative.

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

If the vector is perpendicular to the coordinate axis, then its projection on this axis is equal to zero.
If a vector is co-directed with an axis, then its projection onto this axis is equal to the module of the vector.
If the vector is opposite to the coordinate axis, then its projection on this axis is equal in absolute value to the vector modulus, taken with a minus sign.

2. The most general definition of a projection.

From a right triangle ABD: .

Definition 2. Vector projection on any coordinate axis is called 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 direction of the axis.
If the angle is acute, then the cosine has a positive sign, and the projections are positive. For obtuse angles, the cosine has negative sign, so in such cases the projections onto the axis are negative.
- so for vectors perpendicular to the axis, the projection is zero.

Algebraic vector projection on 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:

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

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

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

Vector projection classification

Types of projections by definition vector projection

1. The geometric projection of the vector AB onto the axis (vector) is called the vector A"B", the beginning of which A’ is the projection of the beginning A onto the axis (vector), and the end B’ is the projection of the end B onto the same axis.
2. The algebraic projection of the vector AB onto the axis (vector) is called the length of the vector A"B" , taken with a + or - sign, depending on whether the vector A"B" has the same direction as the axis (vector).

Types of projections by coordinate system

Vector projection properties

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

Vector projection theorems

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

AC"=AB"+B"C"

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

1. projection onto the OX axis.
2. projection onto the OY axis.
3. projection onto a vector.

Projection onto the OX axis	Projection onto the OY axis	Projection to vector
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 a 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 the vector A'B' is opposite to the direction of the vector NM, then the projection of the vector A'B' has a negative sign.
If the vector AB is parallel to the axis OX, then the projection of the vector A'B' is equal to the modulus 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 modulus of the vector AB.	If the vector AB is parallel to the vector NM, then the projection of the vector A'B' is equal to the modulus of the vector AB.
If the vector AB is perpendicular to the axis OX, then the projection of A'B' is equal to zero (zero-vector).	If the vector AB is perpendicular to the OY axis, then the projection of A'B' is equal to zero (a null vector).	If the vector AB is perpendicular to the vector NM, then the projection of A'B' is equal to zero (a null vector).

1. Question: Can the projection of a vector have a negative sign. Answer: Yes, vector projections 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 a vector coincide with the modulus of the vector. Answer: Yes, it can. In this case, the vectors are parallel (or lie on the same line).
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 o with the OX axis (it is given by the vector a). If OE is a scale unit, then |b|=4, so .

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

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

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

In physics for grade 9 (I.K. Kikoin, A.K. Kikoin, 1999),
a task №5
to chapter " CHAPTER 1. GENERAL INFORMATION ABOUT MOVEMENT».

1. What is called the projection of a vector onto the coordinate axis?

1. The projection of the vector a onto the coordinate axis is the length of the segment between the projections of the beginning and end of the vector a (perpendiculars lowered from these points onto the axis) onto this coordinate axis.

2. How is the displacement vector of the body related to its coordinates?

2. The projections of the displacement vector s on the coordinate axes are equal to the change in the corresponding coordinates of the body.

3. If the coordinate of a point increases over time, then what sign does the projection of the displacement vector onto the coordinate axis have? What if it decreases?

3. If the coordinate of a point increases over time, then the projection of the displacement vector onto the coordinate axis will be positive, because in this case, we will go from the projection of the beginning to the projection of the end of the vector in the direction of the axis itself.

If the coordinate of the point decreases over time, then the projection of the displacement vector on the coordinate axis will be negative, because in this case, we will go from the projection of the beginning to the projection of the end of the vector against the directing axis itself.

4. If the displacement vector is parallel to the X axis, then what is the module of the projection of the vector onto this axis? What about the projection module of the same vector onto the Y-axis?

4. If the displacement vector is parallel to the X axis, then the module of the vector projection on this axis is equal to the module of the vector itself, and its projection on the Y axis is zero.

5. Determine the signs of the projections onto the X axis of the displacement vectors shown in Figure 22. How do the coordinates of the body change during these displacements?

5. In all of the following cases, the Y coordinate of the body does not change, and the X coordinate of the body will change as follows:

a) s 1 ;

the projection of the vector s 1 onto the X axis is negative and modulo equal to the length of the vector s 1 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 1 .

b) s 2 ;

the projection of the vector s 2 onto the X axis is positive and equal in absolute value to the length of the vector s 1 . With such a displacement, the X coordinate of the body will increase by the length of the vector s 2 .

c) s 3 ;

the projection of the vector s 3 onto the X axis is negative and equal in absolute value to the length of the vector s 3 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 3 .

d) s 4 ;

the projection of the vector s 4 onto the X axis is positive and equal in absolute value to the length of the vector s 4 . With such a displacement, the X coordinate of the body will increase by the length of the vector s 4 .

e) s 5 ;

the projection of the vector s 5 onto the X axis is negative and equal in absolute value to the length of the vector s 5 . With such a displacement, the X coordinate of the body will decrease by the length of the vector s 5 .

6. If the distance traveled is large, can the displacement modulus be small?

6. Maybe. This is due to the fact that displacement (displacement vector) is a vector quantity, i.e. is a directed straight line segment connecting the initial position of the body with its subsequent positions. And the final position of the body (regardless of the distance traveled) can be arbitrarily close to the initial position of the body. If the final and initial positions of the body coincide, the displacement modulus will be equal to zero.

7. Why is the displacement vector of a body more important in mechanics than the path it has traveled?

7. The main task of mechanics is to determine the position of the body at any time. Knowing the displacement vector of the body, we can determine the coordinates of the body, i.e. the position of the body at any time, and knowing only the distance traveled, we cannot determine the coordinates of the body, because we do not have information about the direction of movement, but we can only judge the length of the path traveled at a given time.

BASIC CONCEPTS OF VECTOR ALGEBRA

Scalar and vector quantities

From the elementary physics course, it is known that some physical quantities, such as temperature, volume, body mass, density, etc., are determined only by a numerical value. Such quantities are called scalars, or scalars.

To determine some other quantities, such as force, speed, acceleration, and the like, in addition to numerical values, it is also necessary to set their direction in space. Quantities that, in addition to absolute magnitude, are also characterized by direction are called vector.

Definition A vector is a directed segment, which is defined by two points: the first point defines the beginning of the vector, and the second - its end. Therefore, they also say that a vector is an ordered pair of points.

In the figure, the vector is depicted as a straight line segment, on which the arrow marks the direction from the beginning of the vector to its end. For example, fig. 2.1.

If the beginning of the vector coincides with the point , and end with a dot , then the vector is denoted
. In addition, vectors are often denoted by one small letter with an arrow above it. . In books, sometimes the arrow is omitted, then bold type is used to indicate the vector.

Vectors are null vector which has the same start and end. It is denoted or simply .

The distance between the start and end of a vector is called its length, or module. The vector modulus is indicated by two vertical bars on the left:
, or without arrows
or .

Vectors that are parallel to one line are called collinear.

Vectors lying in the same plane or parallel to the same plane are called coplanar.

The null vector is considered collinear to any vector. Its length is 0.

Definition Two vectors
and
are called equal (Fig. 2.2) if they:
1)collinear; 2) co-directed 3) equal in length.

It is written like this:
(2.1)

From the definition of equality of vectors, it follows that with a parallel transfer of a vector, a vector is obtained that is equal to the initial one, therefore the beginning of the vector can be placed at any point in space. Such vectors (in theoretical mechanics, geometry), the beginning of which can be placed at any point in space, are called free. And it is these vectors that we will consider.

Definition Vector system
is called linearly dependent if there are such constants
, among which there is at least one other than zero, and for which equality holds.

Definition An arbitrary three non-coplanar vectors, which are taken in a certain sequence, are called a basis in space.

Definition If a
- basis and vector, then the numbers
are called the coordinates of the vector in this basis.

We will write the vector coordinates in curly brackets after the vector designation. For example,
means that the vector in some chosen basis has a decomposition:
.

From the properties of multiplication of a vector by a number and addition of vectors, an assertion follows regarding linear actions on vectors that are given by coordinates.

In order to find the coordinates of a vector, if the coordinates of its beginning and end are known, it is necessary to subtract the coordinate of the beginning from the corresponding coordinate of its end.

Linear operations on vectors

Linear operations on vectors are the operations of adding (subtracting) vectors and multiplying a vector by a number. Let's consider them.

Definition Vector product per number
is called a vector coinciding in direction with the vector , if
, which has the opposite direction, if
negative. The length of this vector is equal to the product of the length of the vector per modulo number
.

P example . Build Vector
, if
and
(Fig. 2.3).

When a vector is multiplied by a number, its coordinates are multiplied by that number..

Indeed, if , then

Vector product on the
called vector
;
- opposite direction .

Note that a vector whose length is 1 is called single(or ortho).

Using the operation of multiplying a vector by a number, any vector can be expressed in terms of a unit vector of the same direction. Indeed, dividing the vector for its length (i.e. multiplying on the ), we get a unit vector of the same direction as the vector . We will denote it
. Hence it follows that
.

Definition The sum of two vectors and called vector , which comes out of their common origin and is the diagonal of a parallelogram whose sides are vectors and (Fig. 2.4).

.

By definition of equal vectors
that's why
-triangle rule. The triangle rule can be extended to any number of vectors and thus obtain the polygon rule:
is the vector that connects the beginning of the first vector with the end of the last vector (Fig. 2.5).

So, in order to construct the sum vector, it is necessary to attach the beginning of the second to the end of the first vector, to the end of the second to attach the beginning of the third, and so on. Then the sum vector will be the vector that connects the beginning of the first of the vectors with the end of the last.

When vectors are added, their corresponding coordinates are also added

Indeed, if and
,

If the vectors
and are not coplanar, then their sum is a diagonal
a parallelepiped built on these vectors (Fig. 2.6)

,

where

Properties:

- commutativity;

- associativity;

- distributivity with respect to multiplication by a number

.

Those. a vector sum can be transformed according to the same rules as an algebraic one.

DefinitionThe difference of two vectors and is called such a vector , which, when added to the vector gives a vector . Those.
if
. Geometrically represents the second diagonal of the parallelogram built on the vectors and with a common beginning and directed from the end of the vector to the end of the vector (Fig. 2.7).

Projection of a vector onto an axis. Projection Properties

Recall the concept of a number line. A numerical axis is a straight line on which:

direction (→);

reference point (point O);

segment, which is taken as a unit of scale.

Let there be a vector
and axis . From points and let's drop the perpendiculars on the axis . Let's get the points and - point projections and (Fig. 2.8 a).

Definition Vector projection
per axle called the length of the segment
this axis, which is located between the bases of the projections of the beginning and end of the vector
per axle . It is taken with a plus sign if the direction of the segment
coincides with the direction of the projection axis, and with a minus sign if these directions are opposite. Designation:
.

O definition Angle between vector
and axis called the angle , by which it is necessary to turn the axis in the shortest way so that it coincides with the direction of the vector
.

Let's find
:

Figure 2.8 a shows:
.

On fig. 2.8 b): .

The projection of a vector onto an axis is equal to the product of the length of this vector and the cosine of the angle between the vector and the projection axis:
.

Projection Properties:

If a
, then the vectors are called orthogonal

Example . Vectors are given
,
.Then

.

Example. If the beginning of the vector
is at the point
, and end at a point
, then the vector
has coordinates:

O definition Angle between two vectors and called the smallest angle
(Fig. 2.13) between these vectors, reduced to a common beginning .

Angle between vectors and symbolically written like this: .

It follows from the definition that the angle between vectors can vary within
.

If a
, then the vectors are called orthogonal.

.

Definition. The cosines of the angles of a vector with the coordinate axes are called direction cosines of the vector. If the vector
forms angles with the coordinate axes

.

First, let's remember what is coordinate axis, projection of a point onto an axis and coordinates of a point on the axis.

Coordinate axis is a straight line that is given a direction. You can think of it as a vector with an infinitely large modulus.

Coordinate axis denoted by any letter: X, Y, Z, s, t ... Usually, a point is chosen (arbitrarily) on the axis, which is called the origin and, as a rule, denoted by the letter O. Distances to other points of interest to us are measured from this point.

Projection of a point onto an axis- this is the base of the perpendicular dropped from this point to the given axis (Fig. 8). That is, the projection of a point onto the axis is a point.

Point coordinate per axis is a number, the absolute value of which is equal to the length of the segment of the axis (in the selected scale) enclosed between the beginning 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 beginning and with a minus sign if in the opposite direction.

Scalar projection of a vector onto an axis- this is number, the absolute value of which is equal to the length of the segment of the axis (in the selected scale) enclosed between the projections of the start point and the end point of the vector. Important! Usually instead of the expression scalar projection of a vector onto an axis they just say - projection of a vector onto an axis, that is, the word scalar lowered. Vector projection denoted by the same letter as the projected vector (in normal, non-bold writing), with a subscript (usually) 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 a x . When projecting the same vector onto another axis, say the Y axis, its projection will be denoted as y (Fig. 9).

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

and x \u003d x k - x n.

We must remember: the scalar projection of a vector onto an axis (or, simply, the projection of a vector onto an axis) is a number (not a vector)! 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 (Fig. 10).

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 that axis.

Figure 11 shows that a x = a Cos α

That is, the projection of the vector onto the axis is equal to the product of the vector modulus and the cosine of the angle between axis direction and vector direction. If the angle is acute, 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 in the direction - negative. However, since the cosine is an even function, that is, Cos α = Cos (− α), then when calculating projections, the angles can be counted both clockwise and counterclockwise.

When solving problems, the following properties of projections will often be used: if

a = b + c +…+ d, then a x = b x + c x +…+ d x (similarly for other axes),

a= m b, then a x = mb x (similarly for other axes).

The formula a x = a Cos α will be Often meet when solving problems, so it must be known. You need to know the rule for determining the projection by heart!

Remember!

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

Once again - FAST!