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 tasks with a similar theme. 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). From the formula it is clear that from the coordinates of the end point you need to subtract the coordinates starting 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. Faculty students Information technologies They also study the topic of vectors, but at a more complex level.
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 from the coordinates of the end point ($Y$) corresponding coordinates starting point ($X$). We get that
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 down 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
Oxy
ABOUT A OA.
, where
OA
.
Thus, .
Let's look at an example.
Example.
Solution.
:
Answer:
Oxyz in space.
A OA will be a diagonal.
In this case (since OA OA
.
Thus, vector length
.
Example.
Calculate Vector Length
Solution.
, hence,
Answer:
Straight line on a plane
General equation
Ax + By + C ( > 0).
Vector = (A; B) is a normal vector.
In vector form: + C = 0, where is the radius vector of an arbitrary point on a line (Fig. 4.11).
Special cases:
1) By + C = 0- straight line parallel to the axis Ox;
2) Ax + C = 0- straight line parallel to the axis Oy;
3) Ax + By = 0- the straight line passes through the origin;
4) y = 0- axis Ox;
5) x = 0- axis Oy.
Equation of a line in segments
Where a, b- the values of the segments cut off by the straight line on the coordinate axes.
Normal equation of a line(Fig. 4.11)
where is the angle formed normal to the line and the axis Ox; p- the distance from the origin to the straight line.
Bringing general equation straight to normal looking:
Here is the normalized factor of the line; the sign is chosen opposite to the sign C, if and arbitrarily, if C=0.
Finding the length of a vector from coordinates.
We will denote the length of the vector by . Because of this notation, the length of a vector is often called the modulus of the vector.
Let's start by finding the length of a vector on a plane using coordinates.
Let us introduce a rectangular Cartesian coordinate system on the plane Oxy. Let a vector be specified in it and have coordinates . We obtain a formula that allows us to find the length of a vector through the coordinates and .
Let us postpone from the origin of coordinates (from the point ABOUT) vector . Let us denote the projections of the point A on the coordinate axes as and respectively and consider a rectangle with a diagonal OA.
By virtue of the Pythagorean theorem, the equality , where
. From the definition of vector coordinates in a rectangular coordinate system, we can state that and , and by construction the length OA equal to the length of the vector, therefore,
.
Thus, formula for finding the length of a vector according to its coordinates on the plane has the form .
If the vector is represented as a decomposition in coordinate vectors , then its length is calculated using the same formula
, since in this case the coefficients and are the coordinates of the vector in a given coordinate system.
Let's look at an example.
Example.
Find the length of the vector given in Cartesian system coordinates
Solution.
We immediately apply the formula to find the length of the vector from the coordinates :
Answer:
Now we get the formula for finding the length of the vector by its coordinates in a rectangular coordinate system Oxyz in space.
Let us plot the vector from the origin and denote the projections of the point A on the coordinate axes as and . Then we can construct a rectangular parallelepiped on the sides, in which OA will be a diagonal.
In this case (since OA– diagonal rectangular parallelepiped), where . Determining the coordinates of a vector allows us to write equalities, and the length OA equal to the desired vector length, therefore,
.
Thus, vector length in space is equal to the square root of the sum of the squares of its coordinates, that is, found by the formula
.
Example.
Calculate Vector Length , where are the unit vectors of the rectangular coordinate system.
Solution.
We are given a vector decomposition into coordinate vectors of the form , hence,
. Then, using the formula for finding the length of a vector from coordinates, we have .