Find vector length known coordinates of points. Finding the length of the vector coordinates

Find the length of the vector according to its coordinates (in the rectangular coordinate system), according to the coordinates of the beginning and end of the vector and the cosine theorem (2 vector and the angle between them).

Vector - This is the directed cut line.The length of this segment determines the numeric value of the vector and is calledvector length or vector module.

1. Calculation of vector length by its coordinates

If vector coordinates are given in a flat (two-dimensional) rectangular coordinate system, i.e. Known a x and a y, then the length of the vector can be found by the formula

In the case of a vector in space, a third coordinate is added

In MS Excel expression \u003d Root (Summkv (B8: B9)) Allows you to calculate the vector module (it is assumed that the coordinators of the vector are introduced into the cells. B8: B9., See Example File).

The Summkv () function returns the sum of the squares of the arguments, i.e. In this case, the formula is equivalent \u003d B8 * B8 + B9 * B9.

The sample file also calculated the length of the vector in space.

Alternative formula is an expression \u003d Root (Sumpacy (B8: B9; B8: B9)).

2. Finding the length of the vector through the coordinates of the points

If vector set through the coordinates of the points of its beginning and end, then the formula will be another \u003d Root (Summkvson (C28: C29; B28: B29))

The formula assumes that the coordinates of the points of principles and the end are introduced into the ranges C28: C29. and B28: B29. respectively.

Function Summkvson () inpromises the sum of the squares of the differences of the corresponding values \u200b\u200bin two arrays.

In fact, in the formula, the coordinates of the vector (differences of the corresponding points of points) are calculated, then the sum of their squares is calculated.

3. Finding the length of the vector on the cosine theorem

If you want to find the length of the vector on the cosine theorem, then 2 versions are usually given (their modules and angle between them).

Find vector length with using the formula \u003d Root (Summkv (B43: C43) -2 * B43 * C43 * COS (B45))

In cells B43: B43. contains the lengths of vectors a and b, and in the cell B45. - The angle between them in radians (in the fractions of the number Pi ()).

If the angle is given in degrees, then the formula will be slightly different \u003d Root (B43 * B43 + C43 * C43-2 * B43 * C43 * COS (B46 * PI () / 180))

Note: For clarity in the cell with the value of an angle in degrees, you can apply, see for example, an article

First of all, it is necessary to disassemble the very concept of vector. In order to introduce the definition of the geometric vector Recall what segment is. We introduce the following definition.

Definition 1.

Let's call part of the straight line, which has two boundaries in the form of points.

Cut can have 2 directions. To designate the direction, we will call one of the boundaries of the segment of it, and the other border is its end. The direction is indicated from its beginning to the end of the segment.

Definition 2.

A vector or directed segment will be called such a segment for which it is known which of the segment boundaries is considered to be the beginning, and which end it.

Designation: two letters: $ \\ overline (AB) $ - (where $ A $ is its beginning, and $ b $ is its end).

One little letter: $ \\ overline (a) $ (Fig. 1).

We introduce now, directly, the concept of length lengths.

Definition 3.

The vector of the vector $ \\ overline (a) $ will be called the length of the segment of $ a $.

Designation: $ | \\ Overline (a) | $

The concept of the length of the vector is associated, for example, with such a concept as the equality of two vectors.

Definition 4.

Two vectors will be called equal, if they satisfy two conditions: 1. They are coated; 1. Their lengths are equal (Fig. 2).

In order to define the vectors introduce the 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) \u003d M \\ Overline (i) + N \\ Overline (J) $, where $ M $ and $ n $ is valid numbers, and $ \\ Overline (I ) $ and $ \\ overline (j) $ - single vectors on the axis of $ ox $ and $ oy $, respectively.

Definition 5.

The decomposition coefficients of the $ \\ overline (C) \u003d M \\ Overline (I) + N \\ Overline (J) $ will call the coordinates of this vector in the entered coordinate system. Mathematically:

$ \\ Overline (C) \u003d (M, N) $

How to find the length of the vector?

In order to output the formula for calculating the length of an arbitrary vector according to its coordinates, consider the following task:

Example 1.

It is given: vector $ \\ overline (α) $, having coordinates $ (x, y) $. Find: the length of this vector.

We introduce the $ Xoy $ coordinate system on the plane. From the beginning of the introduced coordinate system, I will postpone $ \\ Overline (OA) \u003d \\ Overline (a) $. We construct the projection $ oa_1 $ and $ oa_2 constructed vector on the axis of $ ox $ and $ oy $, respectively (Fig. 3).

The vector of $ \\ overline (oa) $ will be a radius vector for a point $ a $, therefore, it will have coordinates $ (x, y) $, it means

$ \u003d x $, $ [oa_2] \u003d y $

Now we can easily find the desired length using the Pythagora theorem, we get

$ | \\ Overline (α) | ^ 2 \u003d ^ 2 + ^ 2 $

$ | \\ OVERLINE (α) | ^ 2 \u003d x ^ 2 + y ^ 2 $

$ | \\ Overline (α) | \u003d \\ sqrt (x ^ 2 + y ^ 2) $

Answer: $ \\ sqrt (x ^ 2 + y ^ 2) $.

Output:To find the length of the vector that has its coordinates, it is necessary to find the root of the square of the sum of these coordinates.

Example task

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, vector $ \\ overline (XY) $. As we already know, the coordinates of this vector can be found, deducting the coordinates of the end point ($ y $) the corresponding coordinates of the starting point ($ x $). We get that

First of all, it is necessary to disassemble the very concept of vector. In order to introduce the definition of the geometric vector Recall what segment is. We introduce the following definition.

Definition 1.

Let's call part of the straight line, which has two boundaries in the form of points.

Definition 2.

A vector or directed segment will be called such a segment for which it is known which of the segment boundaries is considered to be the beginning, and which end it.

Designation: two letters: $ \\ overline (AB) $ - (where $ A $ is its beginning, and $ b $ is its end).

One little letter: $ \\ overline (a) $ (Fig. 1).

We introduce now, directly, the concept of length lengths.

Definition 3.

The vector of the vector $ \\ overline (a) $ will be called the length of the segment of $ a $.

Designation: $ | \\ Overline (a) | $

The concept of the length of the vector is associated, for example, with such a concept as the equality of two vectors.

Definition 4.

Two vectors will be called equal, if they satisfy two conditions: 1. They are coated; 1. Their lengths are equal (Fig. 2).

Definition 5.

The decomposition coefficients of the $ \\ overline (C) \u003d M \\ Overline (I) + N \\ Overline (J) $ will call the coordinates of this vector in the entered coordinate system. Mathematically:

$ \\ Overline (C) \u003d (M, N) $

How to find the length of the vector?

In order to output the formula for calculating the length of an arbitrary vector according to its coordinates, consider the following task:

Example 1.

It is given: vector $ \\ overline (α) $, having coordinates $ (x, y) $. Find: the length of this vector.

The vector of $ \\ overline (oa) $ will be a radius vector for a point $ a $, therefore, it will have coordinates $ (x, y) $, it means

$ \u003d x $, $ [oa_2] \u003d y $

Now we can easily find the desired length using the Pythagora theorem, we get

$ | \\ Overline (α) | ^ 2 \u003d ^ 2 + ^ 2 $

$ | \\ OVERLINE (α) | ^ 2 \u003d x ^ 2 + y ^ 2 $

$ | \\ Overline (α) | \u003d \\ sqrt (x ^ 2 + y ^ 2) $

Answer: $ \\ sqrt (x ^ 2 + y ^ 2) $.

Output:To find the length of the vector that has its coordinates, it is necessary to find the root of the square of the sum of these coordinates.

Example task

Example 2.

Find the distance between points $ X $ and $ Y $, which have the following coordinates: $ (- 1.5) $ and $ (7.3) $, respectively.

Standard definition: "Vector is a directed segment." Usually, this is limited to the knowledge of the graduate about the vectors. Who needs some "directed segments"?

And in fact, what are the vectors and why they?
Weather forecast. "The wind is northwestern, the speed of 18 meters per second." Agree, the direction of the wind matters (where it blows from), and the module (that is, the absolute value) of its speed.

The values \u200b\u200bthat do not have directions are called scalar. Mass, work, the electric charge is not directed anywhere. They are characterized only numerical meaning - "How many kilograms" or "how much joule".

Physical quantities that have not only absolute value, but also the direction is called vector.

Speed, strength, acceleration - vectors. For them, it is important "how much" and importantly "where". For example, the acceleration of the free fall is directed towards the surface of the Earth, and its value is 9.8 m / s 2. Pulse, electric field strength, induction magnetic field - also vector values.

You remember that physical quantities Denote by letters, Latin or Greek. Arrogo above the letter shows that the value is vector:

Here is another example.
The car moves from A in b. The end result is its movement from point A to point B, that is, moving on a vector .

Now it is clear why the vector is a directed segment. Note, the end of the vector is where the arrow. Length vector Called the length of this segment. Denotes: or

So far, we have worked with scalar values, according to the rules of arithmetic and elementary algebra. Vectors - a new concept. This is another class of mathematical objects. For them, their own rules.

Once we did not know about numbers. Acquaintance with them began in junior classes. It turned out that numbers can be compared with each other, fold, deduct, multiply and divide. We learned that there is a number one and the number zero.
Now we get acquainted with vectors.

The concepts of "more" and "less" for vectors do not exist - they can be different directions. You can only compare the lengths of the vectors.

But the concept of equality for vectors is.
Equal The vectors having the same lengths and the same direction are called. This means that the vector can be transferred in parallel to yourself anywhere in the plane.
Single Called vector, the length of which is equal to 1. Zero - vector, the length of which is zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in a rectangular coordinate system - the very in which draw graphs of functions. Each point in the coordinate system corresponds to two numbers - its coordinates of X and Y, abscissa and ordinate.
The vector also sets two coordinates:

Here in brackets recorded the coordinates of the vector - by x and on y.
They are simply: the coordinate end of the vector minus coordinate of its start.

If the vector coordinates are specified, its length is located by the formula

Addition of vectors

For the addition of vectors there are two ways.

one . Rule parallelogram. To fold the vectors and, we put the start of both at one point. You will be completed to the parallelogram and from the same point we carry out the diagonal of the parallelogram. This will be the sum of vectors and.

Remember the fastener about Swan, Cancer and Pike? They tried very much, but never shifted WHO from the scene. After all, the vector sum of the forces attached to the car was zero.

2. The second way of adding vectors is a triangle rule. Take the same vectors and. By the end of the first vector, I attach the beginning of the second. Now connect the beginning of the first and end of the second. This is the sum of the vectors and.

In the same way, several vectors can be folded. We add them one by one, and then combine the beginning of the first with the end of the latter.

Imagine that you go from point A to paragraph B, from B C, from C in D, then in E and in f. The final result of these actions is moving from a in f.

When adding vectors and get:

Subtract vectors

The vector is sent to the opposite vector. The lengths of the vectors are equal.

Now it is clear what subtraction of vectors. The difference of vectors is the sum of the vector and vector.

Multiplication of vector by number

When vector multiplying the number K, the vector is obtained, the length of which is different from the length. It is coated with a vector if k is larger, and is directed opposite if K is less than zero.

Scalar product vectors

Vectors can be multiplied not only in numbers, but also on each other.

The scalar product of the vectors is the product of the lengths of the vectors on the cosine of the corner between them.

Note - Moved two vectors, and the scalar turned out, that is, the number. For example, in physics, mechanical work is equal to the scalar product of two vectors - forces and movements:

If the vectors are perpendicular, their scalar product is zero.
And here is the scalar product expressed through the coordinates of the vectors and:

From the formula for a scalar product, you can find the angle between vectors:

This formula is especially convenient in stereometry. For example, in the task 14 Profile EME In mathematics, you need to find the angle between cross-go straight or between the straight and plane. Often, the task 14 is solved several times faster than classic.

IN school Program In mathematics, there is only a scalar product of vectors.
It turns out that except for scalar, there is also a vector product when the vector is as a result of multiplying vectors. Who gives the exam in physics, knows what the power of Lorentz and the power of the amper. The formula for finding these forces includes vector art.

Vectors - useful mathematical instrument. In this you will see for the first year.