Pr a b = |b|cos(a,b) or
Where a b is the scalar product of vectors, |a| - modulus of vector a.
Instructions. To find the projection of the vector Pr a b online, you must specify the coordinates of vectors a and b. In this case, the vector can be specified on the 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 you need to use this calculator.
Classification of vector projections
Types of projections by definition vector projection
- 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.
- 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 according to the coordinate system
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 summands of the vectors onto 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
- projection onto the OX axis.
- projection onto the OY axis.
- projection onto a vector.
| Projection on the OX axis | Projection on the OY axis | Projection to vector |
If the direction of vector A’B’ coincides with the direction of the OX axis, then the projection of 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 vector A’B’ coincides with the direction of vector NM, then the projection of 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 vector A’B’ is opposite to the direction of the OY axis, then the projection of 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 vector AB is parallel to the OX axis, then the projection of vector A’B’ is equal to the absolute value of vector AB.   | If vector AB is parallel to the OY axis, then the projection of vector A’B’ is equal to the absolute value of vector AB.   | If vector AB is parallel to vector NM, then the projection of vector A’B’ is equal to the absolute value of vector AB.   |
If the vector AB is perpendicular to the axis OX, then the projection A’B’ is equal to zero (null vector).   | If the vector AB is perpendicular to the OY axis, then the projection A’B’ is equal to zero (null vector).   | If the vector AB is perpendicular to the vector NM, then the projection A’B’ is equal to zero (null vector).   |
1. Question: Can the projection of a vector have a negative sign? Answer: Yes, the projection vector can be a negative value. 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 absolute value 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 (null 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 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 (a,b) = 120 o with the OX axis (with vector a). Length |b| vector b is equal to 4, so 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.
The axis is the direction. This means that projection onto an axis or onto a directed line is considered the same. Projection can be algebraic or geometric. In geometric terms, the projection of a vector onto an axis is understood as a vector, and in algebraic terms, it is understood as a number. That is, the concepts of projection of a vector onto an axis and numerical projection of a vector onto an axis are used.
If we have an L axis and a non-zero vector A B →, then we can construct a vector A 1 B 1 ⇀, denoting the projections of its points A 1 and B 1.
A 1 B → 1 will be the projection of the vector A B → onto L.
Definition 1
Projection of the vector onto the axis is a vector whose beginning and end are projections of the beginning and end beyond given vector. n p L A B → → it is customary to denote the projection A B → onto L. To construct a projection onto L, perpendiculars are dropped onto L.
Example 1
An example of a vector projection onto an axis.
On coordinate plane About x y the point M 1 (x 1 , y 1) is specified. It is necessary to construct projections on O x and O y to image the radius vector of point M 1. We get the coordinates of the vectors (x 1, 0) and (0, y 1).
If we are talking about the projection of a → onto a non-zero b → or the projection of a → onto the direction b → , then we mean the projection of a → onto the axis with which the direction b → coincides. The projection of a → onto the line defined by b → is designated n p b → a → → . It is known that when the angle between a → and b → , n p b → a → → and b → can be considered codirectional. In the case where the angle is obtuse, n p b → a → → and b → are in opposite directions. In a situation of perpendicularity a → and b →, and a → is zero, the projection of a → in the direction b → is a zero vector.
The numerical characteristic of the projection of a vector onto an axis is the numerical projection of a vector onto a given axis.
Definition 2
Numerical projection of the vector onto the axis is a number that is equal to the product of the length of a given vector and the cosine of the angle between the given vector and the vector that determines the direction of the axis.
The numerical projection of A B → onto L is denoted n p L A B → , and a → onto b → - n p b → a → .
Based on the formula, we obtain n p b → a → = a → · cos a → , b → ^ , from where a → is the length of the vector a → , a ⇀ , b → ^ is the angle between the vectors a → and b → .
We obtain the formula for calculating the numerical projection: n p b → a → = a → · cos a → , b → ^ . It is applicable for known lengths a → and b → and the angle between them. The formula is applicable for known coordinates a → and b →, but there is a simplified form.
Example 2
Find out the numerical projection of a → onto a straight line in the direction b → with a length a → equal to 8 and an angle between them of 60 degrees. By condition we have a ⇀ = 8, a ⇀, b → ^ = 60 °. This means that we substitute the numerical values into the formula n p b ⇀ a → = a → · cos a → , b → ^ = 8 · cos 60 ° = 8 · 1 2 = 4 .
Answer: 4.
With known cos (a → , b → ^) = a ⇀ , b → a → · b → , we have a → , b → as dot product a → and b → . Following from the formula n p b → a → = a → · cos a ⇀ , b → ^ , we can find the numerical projection a → directed along the vector b → and get n p b → a → = a → , b → b → . The formula is equivalent to the definition given at the beginning of the paragraph.
Definition 3
The numerical projection of the vector a → onto an axis coinciding in direction with b → is the ratio of the scalar product of the vectors a → and b → to the length b → . The formula n p b → a → = a → , b → b → is applicable to find the numerical projection of a → onto a line coinciding in direction with b → , with known a → and b → coordinates.
Example 3
Given b → = (- 3 , 4) . Find the numerical projection a → = (1, 7) onto L.
Solution
On the coordinate plane n p b → a → = a → , b → b → has the form n p b → a → = a → , b → b → = a x b x + a y b y b x 2 + b y 2 , with a → = (a x , a y ) and b → = b x , b y . To find the numerical projection of vector a → onto the L axis, you need: n p L a → = n p b → a → = a → , b → b → = a x · b x + a y · b y b x 2 + b y 2 = 1 · (- 3) + 7 · 4 (- 3) 2 + 4 2 = 5.
Answer: 5.
Example 4
Find the projection of a → on L, coinciding with the direction b →, where there are a → = - 2, 3, 1 and b → = (3, - 2, 6). Three-dimensional space is specified.
Solution
Given a → = a x , a y , a z and b → = b x , b y , b z , we calculate the scalar product: a ⇀ , b → = a x · b x + a y · b y + a z · b z . The length b → is found using the formula b → = b x 2 + b y 2 + b z 2 . It follows that the formula for determining the numerical projection a → will be: n p b → a ⇀ = a → , b → b → = a x · b x + a y · b y + a z · b z b x 2 + b y 2 + b z 2 .
Substitute the numerical values: n p L a → = n p b → a → = (- 2) 3 + 3 (- 2) + 1 6 3 2 + (- 2) 2 + 6 2 = - 6 49 = - 6 7 .
Answer: - 6 7.
Let's look at the connection between a → on L and the length of the projection a → on L. Let's draw an axis L, adding a → and b → from a point on L, after which we draw a perpendicular line from the end a → to L and draw a projection onto L. There are 5 variations of the image:
First the case with a → = n p b → a → → means a → = n p b → a → → , hence n p b → a → = a → · cos (a , → b → ^) = a → · cos 0 ° = a → = n p b → a → → .
Second the case implies the use of n p b → a → ⇀ = a → · cos a → , b → , which means n p b → a → = a → · cos (a → , b →) ^ = n p b → a → → .
Third the case explains that when n p b → a → → = 0 → we obtain n p b ⇀ a → = a → · cos (a → , b → ^) = a → · cos 90 ° = 0 , then n p b → a → → = 0 and n p b → a → = 0 = n p b → a → → .
Fourth the case shows n p b → a → → = a → · cos (180 ° - a → , b → ^) = - a → · cos (a → , b → ^) , follows n p b → a → = a → · cos (a → , b → ^) = - n p b → a → → .
Fifth the case shows a → = n p b → a → → , which means a → = n p b → a → → , hence we have n p b → a → = a → · cos a → , b → ^ = a → · cos 180° = - a → = - n p b → a → .
Definition 4
The numerical projection of the vector a → onto the L axis, which is directed in the same way as b →, has the following value:
- the length of the projection of the vector a → onto L, provided that the angle between a → and b → is less than 90 degrees or equal to 0: n p b → a → = n p b → a → → with the condition 0 ≤ (a → , b →) ^< 90 ° ;
- zero provided that a → and b → are perpendicular: n p b → a → = 0, when (a → , b → ^) = 90 °;
- projection length a → onto L, multiplied by -1, when there is an obtuse or rotated angle of the vectors a → and b →: n p b → a → = - n p b → a → → with the condition of 90 °< a → , b → ^ ≤ 180 ° .
Example 5
Given the length of the projection a → onto L, equal to 2. Find the numerical projection a → provided that the angle is 5 π 6 radians.
Solution
From the condition it is clear that this angle is obtuse: π 2< 5 π 6 < π . Тогда можем найти числовую проекцию a → на L: n p L a → = - n p L a → → = - 2 .
Answer: - 2.
Example 6
Given a plane O x y z with a vector length a → equal to 6 3, b → (- 2, 1, 2) with an angle of 30 degrees. Find the coordinates of the projection a → onto the L axis.
Solution
First, we calculate the numerical projection of the vector a →: n p L a → = n p b → a → = a → · cos (a → , b →) ^ = 6 3 · cos 30 ° = 6 3 · 3 2 = 9 .
By condition, the angle is acute, then the numerical projection a → = the length of the projection of the vector a →: n p L a → = n p L a → → = 9. This case shows that the vectors n p L a → → and b → are co-directed, which means there is a number t for which the equality is true: n p L a → → = t · b → . From here we see that n p L a → → = t · b → , which means we can find the value of the parameter t: t = n p L a → → b → = 9 (- 2) 2 + 1 2 + 2 2 = 9 9 = 3 .
Then n p L a → → = 3 · b → with the coordinates of the projection of vector a → onto the L axis equal to b → = (- 2 , 1 , 2) , where it is necessary to multiply the values by 3. We have n p L a → → = (- 6 , 3 , 6) . Answer: (- 6, 3, 6).
It is necessary to repeat the previously learned information about the condition of collinearity of vectors.
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First, let's remember what it is coordinate axis, projection of a point onto an axis And coordinates of a point on the axis.
Coordinate axis- This is a straight line that is given some direction. You can think of it as a vector with an infinitely large modulus.
Coordinate axis denoted 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, denoted by the letter O. From this point the distances to other points of interest to us are measured.
Projection of a point onto an axis- this is the base of the perpendicular lowered from this point to this axis (Fig. 8). That is, the projection of a point onto the axis is a point.
Point coordinate on axis- this is the number absolute value which is equal to the length of the axis segment (on the selected scale) enclosed 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.
Scalar projection of a vector onto an axis- This 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. Important! Usually instead of the expression scalar projection of a vector onto an axis they simply say - projection of the vector onto the axis, that is, the word scalar lowered. Vector projection is denoted by the same letter as the projected vector (in normal, non-bold writing), with a lower (as a rule) index 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, say, the Y axis, its projection will be denoted a y (Fig. 9).
To calculate projection of the vector onto the 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.
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 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 this axis.
From Figure 11 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 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 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.
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 (similar to other axes),
a= m b, then a x = mb x (similarly for other axes).
The formula a x = a Cos α will be very often occur when solving problems, so you definitely need to know it. You need to know the rule for determining projection by heart!
Remember!
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.
Once again - by heart!
Solving problems on the equilibrium of converging forces by constructing closed force polygons involves cumbersome constructions. A universal method for solving such problems is to move on to determining the projections of given forces onto the coordinate axes and operating with these projections. An axis is a straight line that is assigned a specific direction.
The projection of a vector onto an axis is a scalar quantity, which is determined by the segment of the axis cut off by the perpendiculars dropped onto it from the beginning and end of the vector.
A vector projection is considered positive if the direction from the beginning of the projection to its end coincides with the positive direction of the axis. A vector projection is considered negative if the direction from the beginning of the projection to its end is opposite to the positive direction of the axis.
Thus, the projection of the force onto the coordinate axis is equal to the product of the force modulus and the cosine of the angle between the force vector and the positive direction of the axis.
Let's consider a number of cases of projecting forces onto an axis:
Force vector F(Fig. 15) is with the positive direction of the x axis acute angle.
To find the projection, from the beginning and end of the force vector we lower perpendiculars to the axis oh; we get
1. F x = F cos α
The projection of the vector in this case is positive
Strength F(Fig. 16) is with the positive direction of the axis X obtuse angle α.
Then F x = F cos α, but since α = 180 0 - φ,
F x = F cos α = F cos180 0 - φ =- F cos φ.
Projection of force F per axis oh in this case it is negative.
Strength F(Fig. 17) perpendicular to the axis oh.
Projection of force F onto the axis X equal to zero
F x = F cos 90° = 0.
Force located on the plane howe(Fig. 18), can be projected on two coordinate axes Oh And oh.
Strength F can be broken down into components: F x and F y. Vector module F x is equal to the projection of the vector F per axis ox, and the vector modulus F y is equal to the projection of the vector F per axis oh.
From Δ OAV: F x = F cos α, F x = F sin α.
From Δ OAS: F x = F cos φ, F x = F sin φ.
The magnitude of the force can be found using the Pythagorean theorem:
The projection of a vector sum or resultant onto any axis is equal to the algebraic sum of the projections of the summands of the vectors onto the same axis.
Consider the converging forces F 1 , F 2 , F 3, and F 4, (Fig. 19, a). The geometric sum, or resultant, of these forces F determined by the closing side of the force polygon
Let us drop from the vertices of the force polygon to the axis x perpendiculars.
Considering the obtained projections of forces directly from the completed construction, we have
F= F 1x+ F 2x+ F 3x+ F 4x
where n is the number of vector terms. Their projections enter the above equation with the corresponding sign.
In a plane, the geometric sum of forces can be projected onto two coordinate axes, and in space, respectively, onto three.
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 that 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 studies and are only expressed in the language of mathematics. However, there is no other language to construct physical theories does not exist.
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 module as the vector c, but is directed in the opposite direction.
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 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.
