First of all, we need to understand the concept of a vector itself. To introduce the definition geometric vector Let's 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
Yandex.RTB R-A-339285-1The length of the vector a → will be denoted by a → . This notation is similar to the modulus of a number, so the length of a vector is also called the modulus of a vector.
To find the length of a vector on a plane from its coordinates, it is necessary to consider a rectangular Cartesian coordinate system O x y. Let some vector a → with coordinates a x be specified in it; ay. Let us introduce a formula for finding the length (modulus) of the vector a → through the coordinates a x and a y.
Let us plot the vector O A → = a → from the origin. Let us determine the corresponding projections of point A onto coordinate axes as A x and A y . Now consider a rectangle O A x A A y with diagonal O A .
From the Pythagorean theorem follows the equality O A 2 = O A x 2 + O A y 2 , whence O A = O A x 2 + O A y 2 . From already known definition vector coordinates in rectangular Cartesian system coordinates we find that O A x 2 = a x 2 and O A y 2 = a y 2 , and by construction, the length of O A is equal to the length of the vector O A →, which means O A → = O A x 2 + O A y 2.
From this it turns out that formula for finding the length of a vector a → = a x ; a y has the corresponding form: a → = a x 2 + a y 2 .
If the vector a → is given as an expansion in coordinate vectors a → = a x · i → + a y · j →, then its length can be calculated using the same formula a → = a x 2 + a y 2, in in this case coefficients a x and a y act as coordinates of the vector a → in given system coordinates
Example 1
Calculate the length of the vector a → = 7 ; e given in rectangular system coordinates
Solution
To find the length of a vector, we will use the formula for finding the length of a vector from coordinates a → = a x 2 + a y 2: a → = 7 2 + e 2 = 49 + e
Answer: a → = 49 + e.
Formula for finding the length of a vector a → = a x ; a y; a z from its coordinates in the Cartesian coordinate system Oxyz in space, is derived similarly to the formula for the case on a plane (see figure below)
In this case, O A 2 = O A x 2 + O A y 2 + O A z 2 (since OA is a diagonal rectangular parallelepiped), hence O A = O A x 2 + O A y 2 + O A z 2 . From the definition of vector coordinates we can write the following equalities O A x = a x ; O A y = a y ; O A z = a z ; , and the length OA is equal to the length of the vector that we are looking for, therefore, O A → = O A x 2 + O A y 2 + O A z 2 .
It follows that the length of the vector a → = a x ; a y; a z is equal to a → = a x 2 + a y 2 + a z 2 .
Example 2
Calculate the length of the vector a → = 4 · i → - 3 · j → + 5 · k → , where i → , j → , k → are the unit vectors of the rectangular coordinate system.
Solution
The vector decomposition a → = 4 · i → - 3 · j → + 5 · k → is given, its coordinates are a → = 4, - 3, 5. Using the above formula we get a → = a x 2 + a y 2 + a z 2 = 4 2 + (- 3) 2 + 5 2 = 5 2.
Answer: a → = 5 2 .
Length of a vector through the coordinates of its start and end points
Formulas were derived above that allow you to find the length of a vector from its coordinates. We considered cases on the plane and in three-dimensional space. Let's use them to find the coordinates of a vector from the coordinates of its start and end points.
So, given the points with given coordinates A (a x ; a y) and B (b x ; b y), hence the vector A B → has coordinates (b x - a x ; b y - a y) which means its length can be determined by the formula: A B → = (b x - a x) 2 + ( b y - a y) 2
And if points with given coordinates A (a x ; a y ; a z) and B (b x ; b y ; b z) are given in three-dimensional space, then the length of the vector A B → can be calculated using the formula
A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2
Example 3
Find the length of the vector A B → if in the rectangular coordinate system A 1, 3, B - 3, 1.
Solution
Using the formula for finding the length of a vector from the coordinates of the start and end points on the plane, we obtain A B → = (b x - a x) 2 + (b y - a y) 2: A B → = (- 3 - 1) 2 + (1 - 3) 2 = 20 - 2 3 .
The second solution involves applying these formulas in turn: A B → = (- 3 - 1 ; 1 - 3) = (- 4 ; 1 - 3) ; A B → = (- 4) 2 + (1 - 3) 2 = 20 - 2 3 . -
Answer: A B → = 20 - 2 3 .
Example 4
Determine at what values the length of the vector A B → is equal to 30 if A (0, 1, 2); B (5 , 2 , λ 2) .
Solution
First, let's write down the length of the vector A B → using the formula: A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2 = (5 - 0) 2 + (2 - 1) 2 + (λ 2 - 2) 2 = 26 + (λ 2 - 2) 2
Then we equate the resulting expression to 30, from here we find the required λ:
26 + (λ 2 - 2) 2 = 30 26 + (λ 2 - 2) 2 = 30 (λ 2 - 2) 2 = 4 λ 2 - 2 = 2 and λ 2 - 2 = - 2 λ 1 = - 2, λ 2 = 2, λ 3 = 0.
Answer: λ 1 = - 2, λ 2 = 2, λ 3 = 0.
Finding the length of a vector using the cosine theorem
Alas, in problems the coordinates of the vector are not always known, so we will consider other ways to find the length of the vector.
Let the lengths of two vectors A B → , A C → and the angle between them (or the cosine of the angle) be given, and you need to find the length of the vector B C → or C B → . In this case, you should use the cosine theorem in the triangle △ A B C and calculate the length of the side B C, which is equal to the desired length of the vector.
Let's consider this case using the following example.
Example 5
The lengths of the vectors A B → and A C → are 3 and 7, respectively, and the angle between them is π 3. Calculate the length of the vector B C → .
Solution
The length of the vector B C → in this case is equal to the length of the side B C of the triangle △ A B C . The lengths of the sides A B and A C of the triangle are known from the condition (they are equal to the lengths of the corresponding vectors), the angle between them is also known, so we can use the cosine theorem: B C 2 = A B 2 + A C 2 - 2 A B A C cos ∠ (A B, → A C →) = 3 2 + 7 2 - 2 · 3 · 7 · cos π 3 = 37 ⇒ B C = 37 Thus, B C → = 37 .
Answer: B C → = 37 .
So, to find the length of a vector from coordinates, there are the following formulas a → = a x 2 + a y 2 or a → = a x 2 + a y 2 + a z 2 , from the coordinates of the start and end points of the vector A B → = (b x - a x) 2 + ( b y - a y) 2 or A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2, in some cases the cosine theorem should be used.
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Vectors. Actions with vectors. In this article we will talk about what a vector is, how to find its length, and how to multiply a vector by a number, as well as how to find the sum, difference and scalar product two vectors.
As usual, a little of the most necessary theory.
A vector is a directed segment, that is, a segment that has a beginning and an end:
Here point A is the beginning of the vector, and point B is its end.
A vector has two parameters: its length and direction.
The length of a vector is the length of the segment connecting the beginning and end of the vector. The vector length is denoted
Two vectors are said to be equal if they have same length and co-directed.
The two vectors are called co-directed, if they lie on parallel lines and are directed in the same direction: vectors and codirectional:
Two vectors are called oppositely directed if they lie on parallel lines and are directed in opposite directions: vectors and , as well as and are directed in opposite directions:
Vectors lying on parallel lines are called collinear: vectors, and are collinear.
Product of a vector a number is called a vector codirectional to the vector if title="k>0">, и направленный в !} the opposite side, if , and whose length is equal to the length of the vector multiplied by:
To add two vectors and, you need to connect the beginning of the vector to the end of the vector. The sum vector connects the beginning of the vector to the end of the vector:
This vector addition rule is called triangle rule.
To add two vectors by parallelogram rule, you need to postpone the vectors from one point and build them up to a parallelogram. The sum vector connects the starting point of the vectors with opposite angle parallelogram:
Difference of two vectors is determined through the sum: the difference of vectors and is called such a vector, which in sum with the vector will give the vector:
It follows from this rule for finding the difference of two vectors: in order to subtract a vector from a vector, you need to plot these vectors from one point. The difference vector connects the end of the vector to the end of the vector (that is, the end of the subtrahend to the end of the minuend):
To find angle between vector and vector, you need to plot these vectors from one point. The angle formed by the rays on which the vectors lie is called the angle between the vectors:
The scalar product of two vectors is the number equal to the product the lengths of these vectors by the cosine of the angle between them:
I suggest you solve problems from Open Bank tasks for , and then check your solution with VIDEO TUTORIALS:
1 . Task 4 (No. 27709)
Two sides of a rectangle ABCD are equal to 6 and 8. Find the length of the difference between the vectors and .
2. Task 4 (No. 27710)
Two sides of a rectangle ABCD are equal to 6 and 8. Find the scalar product of the vectors and . (drawing from the previous task).
3. Task 4 (No. 27711)
Two sides of a rectangle ABCD O. Find the length of the sum of the vectors and .
4 . Task 4 (No. 27712)
Two sides of a rectangle ABCD are equal to 6 and 8. The diagonals intersect at the point O. Find the length of the difference between the vectors and . (drawing from the previous task).
5 . Task 4 (No. 27713)
Diagonals of a rhombus ABCD are equal to 12 and 16. Find the length of the vector.
6. Task 4 (No. 27714)
Diagonals of a rhombus ABCD are equal to 12 and 16. Find the length of the vector +.
7.Task 4 (No. 27715)
Diagonals of a rhombus ABCD are equal to 12 and 16. Find the length of the vector - .(drawing from the previous problem).
8.Task 4 (No. 27716)
Diagonals of a rhombus ABCD are equal to 12 and 16. Find the length of the vector - .
9 . Task 4 (No. 27717)
Diagonals of a rhombus ABCD intersect at a point O and are equal to 12 and 16. Find the length of the vector + .
10 . Task 4 (No. 27718)
Diagonals of a rhombus ABCD intersect at a point O and are equal to 12 and 16. Find the length of the vector - .(drawing from the previous problem).
11.Task 4 (No. 27719)
Diagonals of a rhombus ABCD intersect at a point O and are equal to 12 and 16. Find the scalar product of the vectors and . (drawing from the previous problem).
12 . Task 4 (No. 27720)
ABC are equal Find the length of the vector +.
13 . Task 4 (No. 27721)
Parties regular triangle ABC are equal to 3. Find the length of the vector -. (drawing from the previous problem).
14 . Task 4 (No. 27722)
Sides of a regular triangle ABC are equal to 3. Find the scalar product of the vectors and . (drawing from the previous task).
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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
Since our school days we have known what it is vector is a segment that has a direction and is characterized by numerical value ordered pair of points. The number equal to the length of the segment that serves as the basis is defined as vector length . To define it we will use coordinate system. We also take into account one more characteristic - direction of the segment . In order to find the length of a vector, you can use two methods. The simplest one is to take a ruler and measure what it will be. Or you can use the formula. We will now consider this option.
Necessary:
— coordinate system (x, y);
— vector;
- knowledge of algebra and geometry.
Instructions:
- Formula for determining the length of a directed segment let's write down in the following way r²= x²+y². Taking the square root of r² and the resulting number will be the result. To find the length of a vector, we perform the following steps. We designate the starting point of coordinates (x1;y1), end point (x2;y2). We find x And y by the difference between the coordinates of the end and the beginning of the directed segment. In other words, the number (X) determined by the following formula x=x2-x1, and the number (y) respectively y=y2-y1.
- Find the square of the sum of coordinates using the formula x²+y². We extract the square root of the resulting number, which will be the length of the vector (r). The solution to the problem will be simplified if the initial data of the coordinates of the directed segment are immediately known. All you need to do is plug the data into the formula.
- Attention! The vector may not be on the coordinate plane, but in space, in which case one more value will be added to the formula, and it will have next view: r²= x²+y²+ z², Where - (z) an additional axis that helps determine the size of a directed segment in space.