Vectors and operations on vectors. Vectors All formulas related to vectors in space

Note... Actions with vectors, carried out in example 2.10, can be performed on a personal computer. To find the dot product of vectors in MS Excel, use the SUMPRODUCT () function, where the addresses of the ranges of matrix elements, the sum of the products of which must be found, are specified as arguments. In MathCAD, the dot product of two vectors is performed using the corresponding operator on the Matrix toolbar

Example 2.11. Calculate the work done by forceif the point of its application moves rectilinearly from the position A(2; 4; 6) to position A(4; 2; 7). At what angle to AB force directed ?

Decision.Find the displacement vector by subtracting from the coordinates of its endstart coordinates

... By formula (2.28) (units of work).

Angle φ between and we find by formula (2.29), that is,

6. Three non-coplanar vectors, taken in the indicated order, formright three, if, when viewed from the end of the third vector shortest turn from first vector to the second vectoris performed counterclockwise, andleft if clockwise.

Vector product vector by vector vector is called satisfying the following conditions:

– perpendicular to vectors and;

- has a length equal towhere φ - angle formed by vectorsand;

- vectors form a right-hand triplet (Fig. 2.15).

Theorem 2.4. A necessary and sufficient condition for collinearity of two vectors is the equality to zero of their vector product

Theorem 2.5. Vector product of vectorsgiven by their coordinates is equal to a third-order determinant of the form

(2.32)

Note.Determinant (2.25) decomposes according to property 7 of the determinants

Corollary 1.A necessary and sufficient condition for collinearity of two vectors is the proportionality of their corresponding coordinates

Corollary 2.The vector products of unit unit vectors are

Corollary 3.The vector square of any vector is zero

Geometric interpretation of a vector product is that the length of the resulting vector is numerically equal to the area S a parallelogram built on vectors-factors as on sides reduced to one origin. Indeed, according to the definition, the modulus of the vector product of vectors is. On the other hand, the area of \u200b\u200ba parallelogram built on vectors and is also equal to ... Consequently,

. (2.33)

Also, using the cross product, you can determine the moment of force relative to the point and the linear rotational speed.

Let at the point A force applied let it go O - some point in space (Fig. 2.16). It is known from the physics course that moment of power relative to point O vector is called that goes through the pointO and satisfies the following conditions:

Perpendicular to the plane passing through the points O, A, B;

Its modulus is numerically equal to the product of force per shoulder.

- forms a right triplet with vectors and.

Therefore, the moment of force relative to pointO is the cross product

. (2.34)

axis point (fig.2.17).

Example 2.12.Find the area of \u200b\u200ba triangle using the cross product ABCbuilt on vectorsbrought to the same beginning.

Standard definition: "A vector is a directional line." This is usually what the graduate knows about vectors is limited to. Who needs some "directional lines"?

But in fact, what are vectors and why are they?
Weather forecast. "North-west wind, speed 18 meters per second." You must admit that both the direction of the wind (where it is blowing from) and the modulus (that is, the absolute value) of its speed matter.

Quantities that have no direction are called scalar. Mass, work, electric charge are not directed anywhere. They are characterized only by a numerical value - "how many kilograms" or "how many joules."

Physical quantities that have not only an absolute value, but also a direction are called vector.

Velocity, force, acceleration are vectors. For them, “how much” is important and “where” is important. For example, the acceleration of gravity is directed towards the surface of the Earth, and its magnitude is 9.8 m / s 2. Impulse, electric field strength, magnetic field induction are also vector quantities.

You remember that physical quantities are denoted by letters, Latin or Greek. The arrow above the letter indicates that the value is vector:

Here's another example.
The car moves from A to B. The end result is its movement from point A to point B, that is, moving by a vector .

Now it's clear why a vector is a directional line. Notice that the end of the vector is where the arrow is. Vector length is the length of this segment. Indicated by: or

So far, we have worked with scalars, according to the rules of arithmetic and elementary algebra. Vectors are a new concept. This is a different class of mathematical objects. They have their own rules.

Once we knew nothing about numbers. Acquaintance with them began in the lower grades. It turned out that numbers can be compared with each other, added, subtracted, multiplied and divided. We learned that there is a number one and a number zero.
Now we are introduced to vectors.

The concept of "more" and "less" for vectors does not exist - after all, their directions can be different. Only vector lengths can be compared.

But the concept of equality for vectors is.
Equal vectors are called having the same length and the same direction. This means that the vector can be transferred parallel to itself to any point in the plane.
Single called a vector whose length is 1. Zero - a vector whose length is zero, that is, its beginning coincides with the end.

It is most convenient to work with vectors in a rectangular coordinate system - the same one in which we draw graphs of functions. Each point in the coordinate system corresponds to two numbers - its x and y coordinates, abscissa and ordinate.
The vector is also specified by two coordinates:

Here, the coordinates of the vector are written in brackets - along x and along y.
They are found simply: the coordinate of the end of the vector minus the coordinate of its beginning.

If the coordinates of the vector are given, its length is found by the formula

Vector addition

There are two ways to add vectors.

one . Parallelogram rule. To add the vectors and, place the origins of both at the same point. We finish building to the parallelogram and draw the diagonal of the parallelogram from the same point. This will be the sum of vectors and.

Remember the fable about the swan, crayfish and pike? They tried very hard, but they did not budge the cart. After all, the vector sum of the forces applied by them to the cart was equal to zero.

2. The second way to add vectors is the triangle rule. Let's take the same vectors and. Attach the beginning of the second to the end of the first vector. Now let's connect the beginning of the first and the end of the second. This is the sum of vectors and.

Several vectors can be added according to the same rule. We attach them one by one, and then we connect the beginning of the first with the end of the last.

Imagine walking from point A to point B, from B to C, from C to D, then to E and to F. The end result of these actions is to move from A to F.

When adding vectors and we get:

Subtracting vectors

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

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

Multiplying a vector by a number

When multiplying a vector by a number k, you get a vector whose length is k times different from its length. It is co-directional with the vector if k is greater than zero, and oppositely directed if k is less than zero.

Dot product of vectors

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

The scalar product of vectors is the product of the lengths of the vectors by the cosine of the angle between them.

Pay attention - we multiplied two vectors, and we got a scalar, that is, a number. For example, in physics, mechanical work is equal to the dot product of two vectors - force and displacement:

If vectors are perpendicular, their dot product is zero.
And this is how the dot product is expressed through the coordinates of the vectors and:

From the dot product formula, you can find the angle between the vectors:

This formula is especially useful in solid geometry. For example, in task 14 of the Profile USE in mathematics, you need to find the angle between crossing straight lines or between a straight line and a plane. Problem 14 is often solved several times faster than the classical one.

In the school curriculum in mathematics, only the dot product of vectors is studied.
It turns out, in addition to the scalar, there is also a cross product, when as a result of multiplying two vectors, a vector is obtained. Those who pass the exam in physics know what the Lorentz force and the Ampere force are. It is the vector products that are included in the formulas for finding these forces.

Vectors are a very useful mathematical tool. You will see this in your first year.

Definition 1.Vector in spacecalled a directed segment.

Thus, vectors, unlike scalars, have two characteristics: length and direction. We will denote vectors by symbols, or and .

(Here ANDand IN- the beginning and end of this vector (Fig. 1)) and IN

The length of the vector is indicated by the modulus symbol: .ANDfig. 1

There are three types of vectors defined by the relation of equality between them:

Anchored vectorsare called equal if their beginnings and ends coincide, respectively. An example of such a vector is a force vector.

Sliding vectorsare called equal if they are located on the same straight line, have the same lengths and directions. An example of such vectors is a velocity vector.

Free or geometric vectorsare considered equal if they can be aligned using parallel transfer.

The Analytical Geometry course covers onlyfree vectors.

Definition 2.A vector whose length is zero is called zerovector, or zero -

vector.

Obviously, the beginning and end of the zero vector are the same. The zero vector has no specific direction or has anydirection.

Definition 3.Two vectors lying on one straight line or parallel lines are called

collinear(fig. 2). Mean:
.a

b

Definition 4.Two collinear and identically directed vectors are called

co-directional.Mean:
.

Now we can give a strict definition of equality of free vectors:

Definition 5.Two free vectors are called equal if they are codirectional and have

the same length.

Definition 6.Three vectors lying in one or parallel planes are called

coplanar.

Two perpendicular vectors are called mutually orthogonal:
.

Definition 7.A vector of unit length is called unit vectoror ortom.

Orth co-directional to a nonzero vector and called unit vectorand :e a .

§2. Linear operations on vectors.

Linear operations are defined on the set of vectors: addition of vectors and multiplication of a vector by a number.

I. Addition of vectors.

The sum of 2 - x vectors is a vector, the beginning of which coincides with the beginning of the first, and the end with the end of the second, provided that the beginning of the second coincides with the end of the first.

L it is easy to see that the sum of two vectors, defined

thus (Fig. 3a), coincides with the sum of vectors,

built according to the parallelogram rule (Fig. 6). b

However, this rule allows you to build a

the sum of any number of vectors (Fig. 3b).

a + b

a

b a + b + c

fig. 3b c

Vectors A vector in space is a directed segment, i.e. a segment in which its beginning and end are indicated. The length, or modulus, of a vector is the length of the corresponding segment. The length of the vectors is denoted respectively. Two vectors are said to be equal if they have the same length and direction. A vector with a beginning at point A and an end at point B is denoted and depicted by an arrow with a beginning at point A and end at point B. Zero vectors are also considered whose beginning coincides with the end. All zero vectors are considered equal to each other. They are designated and their length is considered to be zero.

Addition of vectors The addition operation is defined for vectors. In order to add two vectors and, the vector is set aside so that its beginning coincides with the end of the vector. A vector whose beginning coincides with the beginning of the vector, and the end with the end of the vector is called the sum of vectors and is denoted

Multiplication of a vector by a number The product of a vector by a number t is denoted. By definition, the product of a vector by the number -1 is called the opposite vector and is denoted by definition, a vector has a direction opposite to the vector and the product of a vector by a number t is a vector whose length is equal, and the direction remains the same if t\u003e 0, and changes to the opposite if t 0, and reverses if t

Properties The difference of vectors is called a vector, which is denoted For multiplication of a vector by a number, properties similar to those of multiplication of numbers are valid, namely: Property 1. (combination law). Property 2. (the first distribution law). Property 3. (second distribution law).

Definition

Scalar quantity - a quantity that can be characterized by a number. For example, length, area, mass, temperature, etc.

Vector the directed segment $ \\ overline (A B) $ is called; point $ A $ is the beginning, point $ B $ is the end of the vector (Fig. 1).

A vector is denoted either by two capital letters - its beginning and end: $ \\ overline (A B) $ or by one small letter: $ \\ overline (a) $.

Definition

If the beginning and end of the vector coincide, then such a vector is called zero... Most often, the null vector is denoted as $ \\ overline (0) $.

The vectors are called collinearif they lie either on one straight line or on parallel lines (Fig. 2).

Definition

The two collinear vectors $ \\ overline (a) $ and $ \\ overline (b) $ are called co-directedif their directions coincide: $ \\ overline (a) \\ uparrow \\ uparrow \\ overline (b) $ (Fig. 3, a). The two collinear vectors $ \\ overline (a) $ and $ \\ overline (b) $ are called oppositely directedif their directions are opposite: $ \\ overline (a) \\ uparrow \\ downarrow \\ overline (b) $ (Fig. 3, b).

Definition

The vectors are called coplanarif they are parallel to the same plane or lie in the same plane (Fig. 4).

Two vectors are always coplanar.

Definition

Length (module) vector $ \\ overline (A B) $ is the distance between its beginning and end: $ | \\ overline (A B) | $

Detailed theory about the length of the vector by reference.

The length of the zero vector is zero.

Definition

A vector whose length is equal to one is called unit vector or orthom.

The vectors are called equalif they lie on one or parallel lines; their directions coincide and their lengths are equal.

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