Definition. Two or more vectors that lie on the same line or on a parallel line to this, are called collinear vectors

Two collinear vector may point in either same or opposite direction. But, they cannot be inclined at some angle from each other for sure. They are commonly used not only in mathematics, but in physics, engineering and technology as well.

The word collinear is formed by combining two words - co and linear. The word "co" here refers to "same" and "linear" means "in a line". It means collinear literally means something which remains in the same line.

If two vectors lie along the same line, then they are known as collinear vectors. In mathematical language, any two parallel vectors are also called collinear vectors, since they point in either exactly same or exactly opposite direction. Thus, two vectors are said to be collinear if and only if they are either along same line or are parallel to each other.

Collinear vectors either are directed in the same direction, or in the opposite direction. If the direction of the direction is the same, it is designated so , if the direction of the direction is set in the opposite direction, then denoted . The zero vector is oriented to any vector.

Definition. Nonzero vectors having the same directions and equal lengths are called equal vectors. All zero vectors are considered equal.

And then, if they are equal, then it must be first , secondly , in the third place.

And vectors and are opposite vectors. They will have the right equality for them .

From the definition of the equality of vectors, the following conclusions are drawn:

1. You can copy vectors to any point in space. Therefore, free vectors are considered in analytic geometry;

2. Collinear vectors can be copied into one straight line;

3. Any two vectors can be copied into one plane;

4. For a vector and a point 0 there is only one point M for which it is true . It is said that the point M is drawn by the point vector by measuring the length of the vector from the point 0.

If the vector is drawn from point 0 or the vector is obtained from the parallel displacement of the vector (Figure 2), then the vectors are not different, they are equal and vectors that represent only one vector , constructed from point A and from point 0.

Аddition of vectors

Let any two vectors and in the plane be given.

Definition. Adding the given and vectors gives the vector , which is obtained in this way: from any point A we postulate the vector and from its and we postpone the second vector and by connecting the initial point with the final one and write them as or (Figure 3). Since the points are uniquely determined, the third vector, which is the sum of two vectors, is also determined by one. This method of adding vectors is a triangle method.

This definition is the addition of vectors, even if the points , , are in the plane

(1.1)

shows the correctness of the equation.

If all three points , , belong to one straight line, And also the case when any two of them or even three coincide (4a, b - fig.), then equality (1.1) holds.

If you draw a triangle from Figure 3 to a parallelogram (Figure 5), then

, . Then it turns out .

Here's a new rule for adding vectors: To add the vectors and from any point, you need to measure the vectors , fill them to a parallelogram. The diagonal of the parallelogram from the point A is the total vector and is equal to the sum of the given vectors. This method is called the parallelogram method (parallelogram rule)

The triangular method for vector summation can also be used for any number of two vectors. For example, for to find the sum of vectors , starting with any point, you need to measure the vectors , ,…, , to enter (Fig. 6),...,. Then we get a vector that connects the starting point of the first vector with the final point of the last vector and is directed at it, the resulting vector is equal to the sum of the given vectors

This method of summing vectors is a polygonal method.

We consider the basic properties of addition of vectors.

1⁰ A commutative law holds for all vectors and , that is ,

Proof. The proof of this property follows directly from Figure 5, where the vector is parallelly transferred to the point and , the vector in parallel is transferred to the point A and to the point . Because of this rightly, it will .

2⁰ For any three vectors , , , the law (associative law) is satisfied

Proof. We transfer the given vectors in parallel from any point , and at the end of the vector by parallel transfer we mix the next vector, then we transfer the vector from the end of the second vector (Fig. 7). Then we obtain the following vector by the triangle method:

and

. From here .

3⁰ Equality is satisfied for any vector .

Proof. Suppose that . the second connection is a zero vector, so we write it in its form .. Then by the triangle method we have , Hence.

The sum of any vector and the opposite (-) vector is zero to the vector, ie E

4⁰ The sum of any vector and the opposite (- ) vector is zero vector, i.e.

Proof: If , then . To mean it would . Then, .

Difference of vectors

Let and be two vectors in the plane. The difference vector or difference - of and is a vector such that or . Thus we have

In order to build difference of and vectors () we transfer vectors , to one point (Figure 8), moving them in parallel, we add ends. Then . So

. (1.2)

According to the procedure of forming the sum of vectors by setting the addends one after the other, the equation tallies with the picture below; when the minuend and the subtrahend emanate from a common initial point, their difference vector can be directed from the terminal point of the subtrahend to the terminal point of the minuend. In order to build difference of and vectors ()we add to the vector (- ) (Figure 8).

Now the parallelogram rules can be generalized. If we measure vectors , in such a way that we can deduce from one point and fill it up to a parallelogram (Fig. 8), then the diagonal of the point is the sum of the vectors. The vector that is equal to the second diagonal represents the difference of vectors, i.e. , , .

Theorem. Any two vectors have a difference and this difference is unique.

Proof: The vectors and are given. We will place them in such a way as to move them parallel to themselves (Figure 8). Then it happens. Because or . That's why it always happens.

We now prove that the difference is unique. Consider the opposite. Let the difference be not unique, but two different: and . Then by definition of the difference there must be a and .

It follows from the last equality that .

If we add the equality - to both sides , we get , :

Therefore, the vectors and will be one vector.

Multiplication of a vector by a number

Definition. The multiplication of a non-zero vector by a real number is a vector of length , and for the direction of the resulting vector coincides with the direction of the given vector , if , the direction of the resulting vector is opposite to the direction of the given vector and is denoted as .

The definition implies

a) for any vector and number is valid:

b) When the vector is multiplied by a number, we obtain a collinear vector.

From the definition of multiplication of a vector by a number, the following concepts follow:

1) the product of any vector by zero is a zero vector.

Indeed, if . then by definition, we get that the modulus of the vector must be .

The right-hand side of the equation is zero. Hence, the vector is a zero vector. To mean .

2) The product of the zero vector by any number is zero vector.

Indeed, by definition and there is a modulus of the vector .

The right side is zero. Therefore, the vector is zero. Thus we obtain: .

3) The ratio of any vector to its modulus is equal to the zero vector, with the direction that aligns with itself.

Indeed, let any vector and a unit vector with the same direction with it be given. Then this will by definition be multiplication of the vector by a number . This will be multiplied by a number , then The vector is a unit vector . It is called the unit vector or ortvector of vector and is denoted by . So , it's possible . We obtain the same is unit vector.

Multiplication of a vector by a number has the following properties:

1⁰ For any vector , the following will be true: (the special role of the number 1);

2⁰ For any numbers and a vector , the following is true: (commutative property of the numerical factor);

3⁰ for any number and vector , the following is true: (multiplicative distribution coefficient);

4⁰ For any number and vectors , the following is true: (the distribution property with respect to the sum of the vectors);

The proof of property 1⁰ is derived directly from the definition of the product of a number and a vector.

Indeed, the vector and vector modules are equal to and . Their directions are the same. That's it's by definition and It is clear that these modules are the same and their orientation is the same. That's why .

2⁰ - proof of ownership. If or true or equal , the equality is obvious, since both sides of the equation are zero to vectors. Now let's be , . in the near future . Then the vectors and will lie along the line and be equal to the lengths . And when it is oriented with the vector , it will be in the opposite direction . The way it is . Thus, property 2⁰ is proved. All other properties are proved.

Example 1 Given a rhombus , it will be the intersection of its diagonals. In this equation: а) ; б) ; в) ; г) find the multiplier (figure 9).

а) , for the reason ,. ,

b) , , that is , it will .

c) , , as it happens .

d) , , since this happens , .

Theorem (collinear sign). In order that the vector and the non-zero vector are collinear it is necessary and sufficient that there exist only one real number satisfying the equation

(1.3)

Proof:

Necessity. Let the vector be collinear with a nonzero vector . There are three possible situations: , , .

If , then . that is, then the equation (1.3) is satisfied for .

Sufficiency If equation (1.3) is satisfied for a real number , then by definition multiplication of the vector by a number and by the definition of collinear vectors we obtain that the vector is collinear to the vector .

Suppose that there is one more number satisfying the conditions of the theorem. Then and so on . Because or of this .

And then , so on or .

Example. prove that these vectors and . are collinear.

Decision. We use the properties of vectors and get the correctness . So, the vectors are collinear

1.2 Linear dependence and linear independence of vectors

Let there be given a vector system of vectors. An expression of this type (where any real number) is a linear combination of the given vector system, and the numbers are its coefficients. For example, there is a linear combination of vectors , its coefficients is . A linear combination of a given vector system defines one specific vector that is vector:

. (1.4)

Thus, the vector is a linear combination of the vectors or is decomposed into vectors or linearly expressed in terms of the vectors .

Definition. If there is at least one of them a nonzero number such . and the equality holds

(1.5)

Then, the system of vectors is called linear dependent

From this definition, if the given vector system is linearly dependent, then one of them will be equal to the linear combination of the others.

Indeed, if (1.5) is not equal to , then

will be. On the contrary, it is true that a linear combination of a given system of vectors is a linear dependent system.

Definition. If equation (1.5) is true , then only in this case the system of vectors is linearly independent.

If at least one of the vectors is a zero vector, then it is difficult to see that the vector system is linearly dependent. Indeed, for one, for example, if 6 then (5.2) is true for case , then it is satisfied . Hence, by definition, vectors are linearly dependent. Then there is no zero vector in the system of linear independent vectors.

Theorem 1. In order for two vectors to be linearly dependent, their collinearity is necessary and sufficient.

Consequence. Two non-collinear vectors will be linearly non-independent.

Definition. Vectors belonging to the same plane or vectors belonging to parallel planes refer to the coplanar vectors.

Theorem 2. In order for three vectors to be linearly dependent, their coplanarity is necessary and sufficient.

Consequence. not coplanar three vectors will be linearly independent.

Theorem 3. In space, any four vectors will be linearly dependent.

Expansion of the vector into two noncollinear vectors in the plane. Basis of the plane.

Theorem. Any self-coplanar vector can be decomposed into vectors , that are not mutually collinear, and this will be the only decomposition.

Evidence. , , - vectors are coplanar and, , - Let not collinear vectors. Since these vectors are coplanar, they will be in the same plane if you transfer them in parallel so that their start points coincide in one point. They will lie in the same plane. Therefore, according to Theorem 2, they are linearly dependent.

Consequently, there are real numbers and the equality holds. Thus, the vector is decomposed as follows and . We now prove that this decomposition is unique. Let's take the opposite, We will assume that the vector decomposes as follows and : So it will be . But, according to the hypothesis of the theorem, they and are not collinear. Therefore, they are linearly independent. Therefore, the last equation , must be fulfilled only after this , . Thus, it is not true that vector can be decomposed into two different types. Thus, if , , there is a complanar, , there is no coincidence, then its equality is fulfilled.
Definition A basis of a plane is any two vectors of this plane that can be obtained in a certain order and are not collinear to each other, a plane.

Since there are many such pairs of vectors in the plane, there are infinitely many planes in the plane. The differences between them are in the vectors that form the basis. Vectors forming a basis refer to the basis vectors. If , there are basis vectors, then the first basic vector is , the is the second base vector. Then (, ) and (, ) there are different bases. The base (, ) is given. and this is a small rotation angle counterclockwise, to match the direction. If the directions coincide with the direction counterclockwise, then this direction is considered right, if on the contrary, ie. clockwise-left direction.

the 10-а figure shows the right vector, the 10-б figure shows the left vector.

We use the right basis,

Let a basis (, ) be given on a plane, then any vector of this plane should be decomposed, according to the theorem just proved, only in a unique way onto the basis vectors: .

Definition. The coordinates of the vector with respect to the bases (, ) on the plane are the coefficients of the expansion and . The vector coordinates are written in parentheses. By definition , the first coordinate of the vector is , the second coordinate of the vector - . For example: if so , if will be so .

Vectors in spaces are decomposed into three vectors.
The basis in space.
Theorem. We can decompose any vector of space without coplanar vectors

, , , but it can be decomposed in a single way.
Evidence. Let the vectors , , not coplanar. then they will be linearly independent. It is good that four vectors , , , are linearly dependent by Theorem 3. Therefore, there are numbers that can not be equal to zero at the same time, and numbers must be found, and the equation must be satisfied, i.e. Any vector of space is not coplanar , , , but decomposes into vectors . We now prove this decomposition. For example, vector are classified use vectors , , as follows . These are two classifications of one vector

By the theorem, vectors , , are not coplanar, therefore they are linearly independent vectors. In this regard, the last equality , and only when this is done. , and Thus, can not decompose for , , in different ways, it is wrong. In other words, the equation is uniquely determined.

Definition. A spatial basis is mutually non-coplanar any three vectors.

Since there are many such pairs of vectors in the plane, there are infinitely many planes in the space. The differences between them are in the vectors that form the basis. Vectors forming a basis refer to the basis vectors. If , , there are basis vectors, then the first basic vector is , the is the second base vector, then the the third basic vector is . Then (, , ) and (, , ) there are different bases. The base , , is given (11-Figure). and , after this is a small rotation angle counter clock wise, to match the direction. If the directions coincide with the direction counter clock wise, then this direction is considered right (11а-Figure), if on the contrary, i.e. clock wise-left direction(11б-Figure).

We use the right basis,

Definition. The coordinates of the vector with respect to the bases (, , ) on the space are the coefficients of the expansion . The vector coordinates are written . By definition , the first coordinate of the vector is , the second coordinate of the vector - , third coordinate of the vector is Z

For example, ABCD parallelogram, O is the point of intersection of diagonals, - basis vectors, find the coordinates of the sides vectors of the parallelogram and the diagonals of the parallelogram,

Decision. To solve the problem, we must write the sides and of the parallelogram, respectively, through the expansion of the vectors, the diagonal through the expansion of the vectors. from the triangle Δ ОВС

For example, ABCD parallelogram, O is the point of intersection of diagonals, -basis vectors, find the coordinates of the vectors of the parallelogram and the diagonals of the parallelogram,

Decision. To solve the problem, we must write the sides of the parallelogram, respectively, through the expansion of the vectors, the diagonal through the expansion of the vectors. from the triangle ABC we find:

. Then .

From here .

; ; .

Example 2. Find the coordinates of the basis vector.

Decision. To find the coordinates of the basis (, , ), we must classify the base vectors. It is classified as follows . That's why . In the same way and it will happen , . It is clear that only one (, , ) base is given by coordinates, and vectors are equal to their coordinates and only then are equal to each other

The application of arithmetic operations to vectors with coordinates,

Let the space basis (, , ) be given. Let and it be the coordinates of the basis , . The following assumptions about vector coordinates are true:

1⁰.The sum of two or more vectors is equal to the sum of the corresponding coordinates of these vectors .

Proof. By definition, the vector coordinates will be and :

Then, depending on the properties of the addition of vectors, we have:

By definition:

This rule is valid, if the number of vectors is greater than two
Example 1, if and so .
2⁰Differences of vectors are equal to their difference of corresponding coordinates.