Coordinates and vectors. The Comprehensive Guide (2020)
The abscissa and ordinate axis are called coordinates vector. Vector coordinates are usually indicated in the form (x, y), and the vector itself as: =(x, y).
Formula for determining vector coordinates for two-dimensional problems.
In the case of a two-dimensional problem, a vector with known coordinates of points A(x 1;y 1) And B(x 2 ; y 2 ) can be calculated:
= (x 2 - x 1; y 2 - y 1).
Formula for determining vector coordinates for spatial problems.
In the case of a spatial problem, a vector with known coordinates of points A (x 1;y 1;z 1 ) and B (x 2 ; y 2 ; z 2 ) can be calculated using the formula:
= (x 2 - x 1 ; y 2 - y 1 ; z 2 - z 1 ).
Coordinates provide a comprehensive description of the vector, since it is possible to construct the vector itself using the coordinates. Knowing the coordinates, it is easy to calculate and vector length. (Property 3 below).
Properties of vector coordinates.
1. Any equal vectors in a single coordinate system have equal coordinates.
2. Coordinates collinear vectors proportional. Provided that none of the vectors is zero.
3. The square of the length of any vector is equal to the sum of the squares of its coordinates.
4.During surgery vector multiplication on real number each of its coordinates is multiplied by this number.
5. When adding vectors, we calculate the sum of the corresponding vector coordinates.
6. Scalar product two vectors is equal to the sum of the products of their corresponding coordinates.
12. Length of a vector, length of a segment, angle between vectors, condition of perpendicularity of vectors.
Vector – This is a directed segment connecting two points in space or in a plane. Vectors are usually denoted either by small letters or by starting and ending points. There is usually a dash at the top.
For example, a vector directed from the point A to the point B, can be designated a ,
Zero vector 0 or 0 - This is a vector whose starting and ending points coincide, i.e. A = B. From here, 0 = – 0 .
Vector length (modulus)a is the length of the segment representing it AB, denoted by |a | . In particular, | 0 | = 0.
The vectors are called collinear, if their directed segments lie on parallel lines. Collinear vectors a And b are designated a || b .
Three or more vectors are called coplanar, if they lie in the same plane.
Vector addition. Since vectors are directed segments, then their addition can be performed geometrically. (Algebraic addition of vectors is described below, in the paragraph “Unit orthogonal vectors”). Let's pretend that
a = AB and b = CD,
then the vector __ __
a + b = AB+ CD
is the result of two operations:
a)parallel transfer one of the vectors so that its starting point coincides with the end point of the second vector;
b)geometric addition, i.e. constructing a resulting vector going from the starting point of the fixed vector to the ending point of the transferred vector.
Subtraction of vectors. This operation is reduced to the previous one by replacing the subtrahend vector with its opposite one: a – b =a + (– b ) .
Laws of addition.
I. a + b = b + a (Transitional law).
II. (a + b ) + c = a + (b + c ) (Combinative law).
III. a + 0 = a .
IV. a + (– a ) = 0 .
Laws for multiplying a vector by a number.
I. 1 · a = a , 0 · a = 0 , m· 0 = 0 , (– 1) · a = – a .
II. ma = a m,| ma | = | m | · | a | .
III. m(na ) = (mn)a . (C o m b e t a l
law of multiplication by number).
IV. (m+n) a = ma +na , (DISTRIBUTIONAL
m(a + b ) = ma + mb . law of multiplication by number).
Dot product of vectors. __ __
Angle between non-zero vectors AB And CD– this is the angle formed by the vectors when they are transferred in parallel until the points are aligned A And C. Dot product of vectorsa And b is called a number equal to the product of their lengths and the cosine of the angle between them:
If one of the vectors is zero, then their scalar product, in accordance with the definition, is equal to zero:
(a, 0 ) = ( 0 , b ) = 0 .
If both vectors are non-zero, then the cosine of the angle between them is calculated by the formula:
Scalar product ( a , a ), equal to | a | 2, called scalar square. Vector length a and its scalar square are related by:
Dot product of two vectors:
- positively, if the angle between the vectors spicy;
- negative, if the angle between the vectors blunt.
The scalar product of two non-zero vectors is equal to zero then and only when the angle between them is straight, i.e. when these vectors are perpendicular (orthogonal):
Properties of the scalar product. For any vectors a, b,c and any number m the following relations are valid:
I. (a, b ) = (b, a ) . (Transitional law)
II. (ma, b ) = m(a, b ) .
III.(a+b,c ) = (a, c ) + (b, c ). (Distributive law)
Unit orthogonal vectors. In any rectangular coordinate system you can enter unit pairwise orthogonal vectorsi , j And k associated with coordinate axes: i – with axle X, j – with axle Y And k – with axle Z. According to this definition:
(i , j ) = (i , k ) = (j , k ) = 0,
| i | =| j | =| k | = 1.
Any vector a can be expressed through these vectors in a unique way: a = xi+ yj+ zk . Another form of recording: a = (x, y, z). Here x, y, z - coordinates vector a in this coordinate system. In accordance with the last relation and properties of unit orthogonal vectors i, j , k The scalar product of two vectors can be expressed differently.
Let a = (x, y, z); b = (u, v, w). Then ( a, b ) = xu + yv + zw.
The scalar product of two vectors is equal to the sum of the products of the corresponding coordinates.
Vector length (modulus) a = (x, y, z ) is equal to:
In addition, we now have the opportunity to conduct algebraic operations on vectors, namely, addition and subtraction of vectors can be performed using coordinates:
a+ b = (x + u, y + v, z + w) ;
a – b = (x–u, y– v, z–w) .
Cross product of vectors. Vector artwork [a, b ] vectorsa Andb (in this order) is called a vector:
There is another formula for the length of the vector [ a, b ] :
| [ a, b ] | = | a | | b | sin ( a, b ) ,
i.e. length ( module ) vector product of vectorsa Andb is equal to the product of the lengths (modules) of these vectors and the sine of the angle between them. In other words: length (modulus) of the vector[ a, b ] numerically equal to the area of a parallelogram built on vectors a Andb .
Properties of a vector product.
I. Vector [ a, b ] perpendicular (orthogonal) both vectors a And b .
(Prove it, please!).
II.[ a, b ] = – [b, a ] .
III. [ ma, b ] = m[a, b ] .
IV. [ a+b,c ] = [ a, c ] + [ b, c ] .
V. [ a, [ b,c ] ] = b (a , c ) – c (a, b ) .
VI. [ [ a, b ] , c ] = b (a , c ) – a (b,c ) .
Necessary and sufficient condition for collinearity vectors a = (x, y, z) And b = (u, v, w) :
Necessary and sufficient condition for coplanarity vectors a = (x, y, z), b = (u, v, w) And c = (p, q, r) :
EXAMPLE The vectors are given: a = (1, 2, 3) and b = (– 2 , 0 ,4).
Calculate their dot and cross products and angle
between these vectors.
Solution. Using the appropriate formulas (see above), we obtain:
a). scalar product:
(a, b ) = 1 · (– 2) + 2 · 0 + 3 · 4 = 10 ;
b). vector product:
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Finding the coordinates of a vector is a fairly common condition for many problems in mathematics. The ability to find vector coordinates will help you in other, more complex problems with similar topics. In this article we will look at the formula for finding vector coordinates and several problems.
Finding the coordinates of a vector in a plane
What is a plane? A plane is considered to be a two-dimensional space, a space with two dimensions (the x dimension and the y dimension). For example, paper is flat. The surface of the table is flat. Any non-volumetric figure (square, triangle, trapezoid) is also a plane. Thus, if in the problem statement you need to find the coordinates of a vector that lies on a plane, we immediately remember about x and y. You can find the coordinates of such a vector as follows: Coordinates AB of the vector = (xB – xA; yB – xA). The formula shows that you need to subtract the coordinates of the starting point from the coordinates of the end point.
Example:
- Vector CD has initial (5; 6) and final (7; 8) coordinates.
- Find the coordinates of the vector itself.
- Using the above formula, we get the following expression: CD = (7-5; 8-6) = (2; 2).
- Thus, the coordinates of the CD vector = (2; 2).
- Accordingly, the x coordinate is equal to two, the y coordinate is also two.
Finding the coordinates of a vector in space
What is space? Space is already a three-dimensional dimension, where 3 coordinates are given: x, y, z. If you need to find a vector that lies in space, the formula practically does not change. Only one coordinate is added. To find a vector, you need to subtract the coordinates of the beginning from the end coordinates. AB = (xB – xA; yB – yA; zB – zA)
Example:
- Vector DF has initial (2; 3; 1) and final (1; 5; 2).
- Applying the above formula, we get: Vector coordinates DF = (1-2; 5-3; 2-1) = (-1; 2; 1).
- Remember, the coordinate value can be negative, there is no problem.
How to find vector coordinates online?
If for some reason you don’t want to find the coordinates yourself, you can use an online calculator. To begin, select the vector dimension. The dimension of a vector is responsible for its dimensions. Dimension 3 means that the vector is in space, dimension 2 means that it is on the plane. Next, insert the coordinates of the points into the appropriate fields and the program will determine for you the coordinates of the vector itself. Everything is very simple.
By clicking on the button, the page will automatically scroll down and give you the correct answer along with the solution steps.
It is recommended to study this topic well, because the concept of a vector is found not only in mathematics, but also in physics. Students of the Faculty of Information Technology also study the topic of vectors, but at a more complex level.
1. Definition of a vector. Vector length. Collinearity, coplanarity of vectors.
A vector is a directed segment. The length or modulus of a vector is the length of the corresponding directed segment.
Vector module a denoted by . Vector a is called unit if . Vectors are called collinear if they are parallel to the same line. Vectors are called coplanar if they are parallel to the same plane.
2. Multiplying a vector by a number. Operation properties.
Multiplying a vector by a number gives an oppositely directed vector that is twice as long. Multiplying a vector by a number in coordinate form is done by multiplying all coordinates by this number:
Based on the definition, we obtain an expression for the modulus of the vector multiplied by the number:
Similar to numbers, the operation of adding a vector to itself can be written through multiplication by a number:
And subtraction of vectors can be rewritten through addition and multiplication:
Based on the fact that multiplication by does not change the length of the vector, but only the direction, and taking into account the definition of a vector, we obtain:
3. Addition of vectors, subtraction of vectors.
In coordinate representation, the sum vector is obtained by summing the corresponding coordinates of the terms:
To geometrically construct a sum vector, various rules (methods) are used, but they all give the same result. The use of one or another rule is justified by the problem being solved.
Triangle rule
The triangle rule follows most naturally from the understanding of a vector as a transfer. It is clear that the result of sequentially applying two transfers at a certain point will be the same as applying one transfer at once that corresponds to this rule. To add two vectors according to the rule triangle both of these vectors are transferred parallel to themselves so that the beginning of one of them coincides with the end of the other. Then the sum vector is given by the third side of the resulting triangle, and its beginning coincides with the beginning of the first vector, and its end with the end of the second vector.
This rule can be directly and naturally generalized to the addition of any number of vectors, turning into broken line rule:
Polygon rule
The beginning of the second vector coincides with the end of the first, the beginning of the third with the end of the second, and so on, the sum of the vectors is a vector, with the beginning coinciding with the beginning of the first, and the end coinciding with the end of the th (that is, it is depicted by a directed segment closing the broken line) . Also called the broken line rule.
Parallelogram rule
To add two vectors and according to the rule parallelogram both of these vectors are transferred parallel to themselves so that their origins coincide. Then the sum vector is given by the diagonal of the parallelogram constructed on them, starting from their common origin. (It is easy to see that this diagonal coincides with the third side of the triangle when using the triangle rule).
The parallelogram rule is especially convenient when there is a need to depict the sum vector as immediately applied to the same point to which both terms are applied - that is, to depict all three vectors as having a common origin.
Vector sum modulus
Modulus of the sum of two vectors can be calculated using cosine theorem:
Where is the cosine of the angle between the vectors.
If the vectors are depicted in accordance with the triangle rule and the angle is taken according to the drawing - between the sides of the triangle - which does not coincide with the usual definition of the angle between vectors, and therefore with the angle in the above formula, then the last term acquires a minus sign, which corresponds to the cosine theorem in its direct formulation.
For the sum of an arbitrary number of vectors a similar formula is applicable, in which there are more terms with cosine: one such term exists for each pair of vectors from the summed set. For example, for three vectors the formula looks like this:
Vector subtraction
Two vectors and their difference vector
To obtain the difference in coordinate form, you need to subtract the corresponding coordinates of the vectors:
To obtain a difference vector, the beginnings of the vectors are connected and the beginning of the vector will be the end, and the end will be the end. If we write it using vector points, then.
Vector difference module
Three vectors, as with addition, form a triangle, and the expression for the difference module is similar:
where is the cosine of the angle between the vectors
The difference from the formula for the modulus of the sum is in the sign in front of the cosine; in this case, you need to carefully monitor which angle is taken (the version of the formula for the modulus of the sum with the angle between the sides of a triangle when summing according to the triangle rule does not differ in form from this formula for the modulus of the difference, but you need to have Note that different angles are taken here: in the case of a sum, the angle is taken when the vector is transferred to the end of the vector; when a difference model is sought, the angle between vectors applied to one point is taken; the expression for the modulus of the sum using the same angle as in given expression for the modulus of the difference, differs in the sign in front of the cosine).
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First of all, we need to understand 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 now introduce directly the concept of vector lengths.
Definition 3
The length of the vector $\overline(a)$ will be the length of the segment $a$.
Notation: $|\overline(a)|$
The concept of vector length is associated, for example, with such a concept as the equality of two vectors.
Definition 4
We will call two vectors equal if they satisfy two conditions: 1. They are codirectional; 1. Their lengths are equal (Fig. 2).
In order to define vectors, enter a coordinate system and determine the coordinates for the vector in the entered system. As we know, any vector can be decomposed in the form $\overline(c)=m\overline(i)+n\overline(j)$, where $m$ and $n$ are real numbers, and $\overline(i )$ and $\overline(j)$ are unit vectors on the $Ox$ and $Oy$ axis, respectively.
Definition 5
We will call the expansion coefficients of the vector $\overline(c)=m\overline(i)+n\overline(j)$ the coordinates of this vector in the introduced coordinate system. Mathematically:
$\overline(c)=(m,n)$
How to find the length of a vector?
In order to derive a formula for calculating the length of an arbitrary vector given its coordinates, consider the following problem:
Example 1
Given: vector $\overline(α)$ with coordinates $(x,y)$. Find: the length of this vector.
Let us introduce a Cartesian coordinate system $xOy$ on the plane. Let us set aside $\overline(OA)=\overline(a)$ from the origins of the introduced coordinate system. Let us construct projections $OA_1$ and $OA_2$ of the constructed vector on the $Ox$ and $Oy$ axes, respectively (Fig. 3).
The vector $\overline(OA)$ we have constructed will be the radius vector for point $A$, therefore, it will have coordinates $(x,y)$, which means
$=x$, $[OA_2]=y$
Now we can easily find the required length using the Pythagorean theorem, we get
$|\overline(α)|^2=^2+^2$
$|\overline(α)|^2=x^2+y^2$
$|\overline(α)|=\sqrt(x^2+y^2)$
Answer: $\sqrt(x^2+y^2)$.
Conclusion: To find the length of a vector whose coordinates are given, it is necessary to find the root of the square of the sum of these coordinates.
Sample tasks
Example 2
Find the distance between points $X$ and $Y$, which have the following coordinates: $(-1.5)$ and $(7.3)$, respectively.
Any two points can be easily associated with the concept of a vector. Consider, for example, the vector $\overline(XY)$. As we already know, the coordinates of such a vector can be found by subtracting the corresponding coordinates of the starting point ($X$) from the coordinates of the end point ($Y$). We get that
