The projections of vector a on the coordinate axes are given. Projection of a vector onto an axis. How to find the projection of a vector
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 of 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. A mathematical model is 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 minds a picture of the physical world that most fully reflects its properties and ensures 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 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 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.
Projecting various lines and surfaces onto a plane allows you to build a visual image of objects in the form of a drawing. We will consider rectangular projection, in which the projecting rays are perpendicular to the projection plane. PROJECTION OF A VECTOR ON A PLANE consider the vector = (Fig. 3.22), enclosed between the perpendiculars omitted from its beginning and end.
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Rice. 3.22. Vector projection of a vector onto a plane.
Rice. 3.23. Vector projection of a vector onto an axis.
In vector algebra, it is often necessary to project a vector onto an AXIS, that is, onto a straight line that has a certain orientation. Such design is easy if the vector and the L axis lie in the same plane (Fig. 3.23). However, the task becomes more difficult when this condition is not met. Let's construct a 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 onto an axis
in general.
Through the ends of the vector we draw planes perpendicular to the line L. At the intersection with this 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 a vector projection can be solved more easily if the vector is brought into the same plane as the axis, which can be done since free vectors are considered in vector algebra.
Along with the vector projection, there is also a 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 its opposite value if the vector projection and the L axis have the opposite orientation. We will denote the scalar projection:
Vector and scalar projections are not always strictly terminologically separated in practice. The term “vector projection” is usually used, meaning a scalar projection of a vector. When making a decision, 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 us to calculate 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 and the cosine of the angle between the vector and the axis, that is
(3.5)
Rice. 3.25. Finding vector and scalar
Vector projections onto the L axis
(and the L axis are equally oriented).
PROOF. Let us first carry out constructions that allow us to find the angle G Between the vector and the L axis. To do this, we will construct a straight line MN, parallel to the L axis and passing through point O - the beginning of the vector (Fig. 3.25). The angle will be the desired angle. Let us draw two planes through points A and O, perpendicular to the L axis. We obtain:
Since the L axis and the straight line MN are parallel.
Let us highlight two cases of relative position of the vector and the L axis.
1. Let the vector projection and the L axis be equally oriented (Fig. 3.25). Then the corresponding scalar projection 
.
2. Let and L be oriented in different sides(Fig. 3.26).
Rice. 3.26. Finding the vector and scalar projections of the vector onto the L axis (and the L axis is oriented in opposite directions).
Thus, in both cases the theorem is true.
THEOREM 6. If the origin of the vector is brought to a certain point on the L axis, and this axis is located in the s plane, the vector forms an angle with the vector projection on the s plane, and an angle with the vector projection on the L axis, in addition, the vector projections themselves form an angle with each other , That
and on an axis or some other vector there are the concepts of its geometric projection and numerical (or algebraic) projection. The result of a geometric projection will be a vector, and the result of an algebraic projection will be 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 us remember what a segment is. Let us introduce the following definition.
Definition 1
A segment is a part of a line that has two boundaries in the form of points.
A segment can have 2 directions. To denote 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 directed segment will be a segment for which it is known which of the boundaries of the segment is considered the beginning and which is its end.
Designation: In two letters: $\overline(AB)$ – (where $A$ is its beginning, and $B$ is its end).
In one small letter: $\overline(a)$ (Fig. 1).
Let us introduce a few more concepts related to the concept of a vector.
Definition 3
We will call two non-zero vectors collinear if they lie on the same line or on lines parallel to each other (Fig. 2).
Definition 4
We will call two non-zero vectors codirectional if they satisfy two conditions:
- These vectors are collinear.
- If they are directed in one direction (Fig. 3).
Notation: $\overline(a)\overline(b)$
Definition 5
We will call two non-zero vectors oppositely directed if they satisfy two conditions:
- These vectors are collinear.
- If they are directed in different directions (Fig. 4).
Notation: $\overline(a)↓\overline(d)$
Definition 6
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
Let's move on to determining the equality of two vectors
Definition 7
We will call two vectors equal if they satisfy two conditions:
- They are co-directional;
- 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 point of the vector $A$ is projected onto this axis. We obtain point $A"$ - the beginning of the desired vector. The end point of vector $B$ is projected onto this axis. We obtain point $B"$ - the end of the desired vector. The vector $\overline(A"B")$ will be the desired vector.
Let's consider the problem:
Example 1
Construct a geometric projection $\overline(AB)$ onto the $l$ axis shown in Figure 6.
Let us draw a perpendicular from point $A$ to the axis $l$, we obtain point $A"$ on it. Next, we draw a perpendicular from point $B$ to the axis $l$, we obtain point $B"$ on it (Fig. 7).
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.
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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 of a 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 the coordinate plane O x y, a 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 the 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 scalar 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 the 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 °;
- the length of the projection a → onto L, multiplied by -1, when there is an obtuse or straight 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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Many physical quantities are completely determined by specifying a certain number. These are, for example, volume, mass, density, body temperature, etc. Such quantities are called scalar. Because of this, numbers are sometimes called scalars. But there are also quantities that are determined by specifying not only a number, but also a certain direction. For example, when a body moves, you should indicate not only the speed at which the body is moving, but also the direction of movement. In the same way, when studying the action of any force, it is necessary to indicate not only the value of this force, but also the direction of its action. Such quantities are called vector. To describe them, the concept of a vector was introduced, which turned out to be useful for mathematics.
Vector definition
Any ordered pair of points A to B in space defines directed segment, i.e. a segment along with the direction specified on it. If point A is the first, then it is called the beginning of the directed segment, and point B is its end. The direction of a segment is considered to be the direction from beginning to end.
Definition
A directed segment is called a vector.
We will denote a vector by the symbol \(\overrightarrow(AB) \), with the first letter indicating the beginning of the vector, and the second - its end.
A vector whose beginning and end coincide is called zero and is denoted by \(\vec(0)\) or simply 0.
The distance between the start and end of a vector is called its length and is denoted by \(|\overrightarrow(AB)| \) or \(|\vec(a)| \).
The vectors \(\vec(a) \) and \(\vec(b) \) are called collinear, if they lie on the same line or on parallel lines. Collinear vectors can have the same or opposite directions.
Now we can formulate the important concept of equality of two vectors.
Definition
Vectors \(\vec(a) \) and \(\vec(b) \) are said to be equal (\(\vec(a) = \vec(b) \)) if they are collinear, have the same direction and their lengths are equal .
In Fig. 1 shows unequal vectors on the left and equal vectors \(\vec(a) \) and \(\vec(b) \) on the right. From the definition of equality of vectors it follows that if a given vector is moved parallel to itself, then the result will be a vector equal to the given one. In this regard, vectors in analytical geometry are called free.
Projection of a vector onto an axis
Let the axis \(u\) and some vector \(\overrightarrow(AB)\) be given in space. Let us draw planes perpendicular to the \(u\) axis through points A and B. Let us denote by A" and B" the points of intersection of these planes with the axis (see Figure 2).
The projection of the vector \(\overrightarrow(AB) \) onto the axis \(u\) is the value A"B" of the directed segment A"B" on the axis \(u\). Let us remind you that
\(A"B" = |\overrightarrow(A"B")| \) , if the direction \(\overrightarrow(A"B") \) coincides with the direction of the axis \(u\),
\(A"B" = -|\overrightarrow(A"B")| \) , if the direction \(\overrightarrow(A"B") \) is opposite to the direction of the axis \(u\),
The projection of the vector \(\overrightarrow(AB)\) onto the axis \(u\) is denoted as follows: \(Pr_u \overrightarrow(AB)\).
Theorem
The projection of the vector \(\overrightarrow(AB) \) onto the axis \(u\) is equal to the length of the vector \(\overrightarrow(AB) \) multiplied by the cosine of the angle between the vector \(\overrightarrow(AB) \) and the axis \( u\) , i.e.
Comment
Let \(\overrightarrow(A_1B_1)=\overrightarrow(A_2B_2) \) and some axis \(u\) be specified. Applying the formula of the theorem to each of these vectors, we obtain
Vector projections on coordinate axes
Let a rectangular coordinate system Oxyz and an arbitrary vector \(\overrightarrow(AB)\) be given in space. Let, further, \(X = Pr_u \overrightarrow(AB), \;\; Y = Pr_u \overrightarrow(AB), \;\; Z = Pr_u \overrightarrow(AB) \). The projections of the X, Y, Z vector \(\overrightarrow(AB)\) on the coordinate axes are called coordinates. At the same time they write
\(\overrightarrow(AB) = (X;Y;Z) \)
Theorem
Whatever the two points A(x 1 ; y 1 ; z 1) and B(x 2 ; y 2 ; z 2), the coordinates of the vector \(\overrightarrow(AB) \) are determined by the following formulas:
X = x 2 -x 1 , Y = y 2 -y 1 , Z = z 2 -z 1
Comment
If the vector \(\overrightarrow(AB) \) leaves the origin, i.e. x 2 = x, y 2 = y, z 2 = z, then the coordinates X, Y, Z of the vector \(\overrightarrow(AB) \) are equal to the coordinates of its end:
X = x, Y = y, Z = z.
Direction cosines of a vector
Let an arbitrary vector \(\vec(a) = (X;Y;Z) \); we will assume that \(\vec(a) \) comes out from the origin and does not lie in any coordinate plane. Let us draw planes perpendicular to the axes through point A. Together with the coordinate planes, they form a rectangular parallelepiped, the diagonal of which is the segment OA (see figure).
From elementary geometry it is known that the square of the diagonal length of a rectangular parallelepiped equal to the sum squares of the lengths of its three dimensions. Hence,
\(|OA|^2 = |OA_x|^2 + |OA_y|^2 + |OA_z|^2 \)
But \(|OA| = |\vec(a)|, \;\; |OA_x| = |X|, \;\; |OA_y| = |Y|, \;\;|OA_z| = |Z| \); thus we get
\(|\vec(a)|^2 = X^2 + Y^2 + Z^2 \)
or
\(|\vec(a)| = \sqrt(X^2 + Y^2 + Z^2) \)
This formula expresses the length of an arbitrary vector through its coordinates.
Let us denote by \(\alpha, \; \beta, \; \gamma \) the angles between the vector \(\vec(a) \) and the coordinate axes. From the formulas for the projection of the vector onto the axis and the length of the vector we obtain
\(\cos \alpha = \frac(X)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \beta = \frac(Y)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \gamma = \frac(Z)(\sqrt(X^2 + Y^2 + Z^2)) \)
\(\cos \alpha, \;\; \cos \beta, \;\; \cos \gamma \) are called direction cosines of the vector \(\vec(a) \).
Squaring the left and right sides of each of the previous equalities and summing up the results obtained, we have
\(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \)
those. the sum of the squares of the direction cosines of any vector is equal to one.
Linear operations on vectors and their basic properties
Linear operations on vectors are the operations of adding and subtracting vectors and multiplying vectors by numbers.Addition of two vectors
Let two vectors \(\vec(a) \) and \(\vec(b) \) be given. The sum \(\vec(a) + \vec(b) \) is a vector that goes from the beginning of the vector \(\vec(a) \) to the end of the vector \(\vec(b) \) provided that the vector \(\vec(b) \) is attached to the end of the vector \(\vec(a) \) (see figure).Comment
The action of subtracting vectors is inverse to the action of addition, i.e. the difference \(\vec(b) - \vec(a) \) vectors \(\vec(b) \) and \(\vec(a) \) is a vector that, in sum with the vector \(\vec(a ) \) gives the vector \(\vec(b) \) (see figure).
Comment
By determining the sum of two vectors, you can find the sum of any number of given vectors. Let, for example, be given three vectors \(\vec(a),\;\; \vec(b), \;\; \vec(c) \). Adding \(\vec(a) \) and \(\vec(b) \), we obtain the vector \(\vec(a) + \vec(b) \). Now adding to it the vector \(\vec(c) \), we obtain the vector \(\vec(a) + \vec(b) + \vec(c) \) 
Product of a vector and a number
Let the vector \(\vec(a) \neq \vec(0) \) and the number \(\lambda \neq 0 \) be given. The product \(\lambda \vec(a) \) is a vector that is collinear to the vector \(\vec(a) \), has length equal to \(|\lambda| |\vec(a)| \), and direction the same as the vector \(\vec(a) \) if \(\lambda > 0 \), and the opposite if \(\lambda Geometric meaning the operations of multiplying the vector \(\vec(a) \neq \vec(0) \) by the number \(\lambda \neq 0 \) can be expressed as follows: if \(|\lambda| >1 \), then when multiplying vector \(\vec(a) \) by the number \(\lambda \) the vector \(\vec(a) \) is “stretched” \(\lambda \) times, and if \(|\lambda| 1 \ ).If \(\lambda =0 \) or \(\vec(a) = \vec(0) \), then the product \(\lambda \vec(a) \) is considered equal to the zero vector.
Comment
Using the definition of multiplying a vector by a number, it is easy to prove that if the vectors \(\vec(a) \) and \(\vec(b) \) are collinear and \(\vec(a) \neq \vec(0) \), then there exists (and only one) number \(\lambda \) such that \(\vec(b) = \lambda \vec(a) \)
Basic properties of linear operations
1. Commutative property of addition
\(\vec(a) + \vec(b) = \vec(b) + \vec(a) \)
2. Combinative property of addition
\((\vec(a) + \vec(b))+ \vec(c) = \vec(a) + (\vec(b)+ \vec(c)) \)
3. Combinative property of multiplication
\(\lambda (\mu \vec(a)) = (\lambda \mu) \vec(a) \)
4. Distributive property regarding the sum of numbers
\((\lambda +\mu) \vec(a) = \lambda \vec(a) + \mu \vec(a) \)
5. Distributive property with respect to the sum of vectors
\(\lambda (\vec(a)+\vec(b)) = \lambda \vec(a) + \lambda \vec(b) \)
Comment
These properties of linear operations are of fundamental importance, since they make it possible to perform ordinary algebraic operations on vectors. For example, due to properties 4 and 5, you can multiply a scalar polynomial by a vector polynomial “term by term”.
Vector projection theorems
Theorem
The projection of the sum of two vectors onto an axis is equal to the sum of their projections onto this axis, i.e.
\(Pr_u (\vec(a) + \vec(b)) = Pr_u \vec(a) + Pr_u \vec(b) \)
The theorem can be generalized to the case of any number of terms.
Theorem
When the vector \(\vec(a) \) is multiplied by the number \(\lambda \), its projection onto the axis is also multiplied by this number, i.e. \(Pr_u \lambda \vec(a) = \lambda Pr_u \vec(a) \)
Consequence
If \(\vec(a) = (x_1;y_1;z_1) \) and \(\vec(b) = (x_2;y_2;z_2) \), then
\(\vec(a) + \vec(b) = (x_1+x_2; \; y_1+y_2; \; z_1+z_2) \)
Consequence
If \(\vec(a) = (x;y;z) \), then \(\lambda \vec(a) = (\lambda x; \; \lambda y; \; \lambda z) \) for any number \(\lambda \)
From here it is easy to deduce condition of collinearity of two vectors in coordinates.
Indeed, the equality \(\vec(b) = \lambda \vec(a) \) is equivalent to the equalities \(x_2 = \lambda x_1, \; y_2 = \lambda y_1, \; z_2 = \lambda z_1 \) or
\(\frac(x_2)(x_1) = \frac(y_2)(y_1) = \frac(z_2)(z_1) \) i.e. the vectors \(\vec(a) \) and \(\vec(b) \) are collinear if and only if their coordinates are proportional.
Decomposition of a vector into a basis
Let the vectors \(\vec(i), \; \vec(j), \; \vec(k) \) be the unit vectors of the coordinate axes, i.e. \(|\vec(i)| = |\vec(j)| = |\vec(k)| = 1 \), and each of them is equally directed with the corresponding coordinate axis (see figure). A triple of vectors \(\vec(i), \; \vec(j), \; \vec(k) \) is called basis.
The following theorem holds.
Theorem
Any vector \(\vec(a) \) can be uniquely expanded over the basis \(\vec(i), \; \vec(j), \; \vec(k)\; \), i.e. presented as
\(\vec(a) = \lambda \vec(i) + \mu \vec(j) + \nu \vec(k) \)
where \(\lambda, \;\; \mu, \;\; \nu \) are some numbers. 
