Vectors.
We will call a directed segment a vector.
The direction of a vector is determined by indicating its beginning and end. In the drawing, the direction of the vector is indicated by an arrow. To denote vectors, we will use urgent Latin letters a, b, c, …. You can also denote a vector by indicating its beginning and end. In this case, the beginning of the vector is placed in first place. Instead of the word ''vector'' above letter designation Vectors are sometimes marked with an arrow or a line.
Vectors and are said to be equally directed if the half-lines AB and CD are equally directed. Vectors and are said to be oppositely directed if the half-lines AB and CD are oppositely directed. Vectors and are equally directed, and vectors and are oppositely directed.
The absolute value (or modulus) of a vector is the length of the segment representing the vector. The absolute value of the vector is denoted by .
The beginning of a vector can coincide with its end. We will call such a vector the zero vector. The zero vector is denoted by . They don't talk about the direction of the zero vector. The absolute value of the zero vector is considered equal to zero.
Equality of vectors
Two vectors are said to be equal if they are combined by parallel translation. This means that there is parallel transfer, which translates the beginning and end of one vector, respectively, into the beginning and end of another vector.
From this definition Equality of vectors implies that equal vectors have the same direction and are equal in absolute value.
Conversely: if the vectors are identically directed and equal in absolute value, then they are equal.
Indeed, even if they are identically directed vectors, equal in absolute value. A parallel translation that moves point C to point A combines the half-line CD with the half-line AB, since they have the same direction.
And since the segments AB and CD are equal, then point D is combined with point B, i.e. parallel translation transforms the vector into a vector. This means that the vectors and are equal, which is what needed to be proven.
Vector coordinates
Let a vector have a beginning at a point and an end at a point. The coordinates of the vector will be the numbers , .
We will place the coordinates of the vector next to the letter designation of the vector, in this case () or simply ( The coordinates of the zero vector are equal to zero.
From the formula expressing the distance between two points through their coordinates, it follows that the absolute value of the vector with coordinates is equal to
Equal vectors have equal corresponding coordinates. And vice versa: if the corresponding coordinates of the vectors are equal, then the vectors are equal.
Indeed, let , and the beginning and end of the vector . Since a vector equal to it receives their vectors by parallel transfer, then its beginning and end will be, respectively 
, 
.
This shows that both vectors have the same coordinates: 
.
Vector addition.
The sum of a vector with coordinates is a vector with coordinates, i.e.
For any vectors , ,
To prove it, it is enough to compare the corresponding coordinates of the vectors on the right and left sides of the equalities. We see that they are equal. And vectors with correspondingly equal coordinates are equal.
Theorem. Whatever the points A, B, C, there is a vector equality
Proof. Let the data be points. A vector has coordinates, a vector has coordinates. Therefore, a vector has coordinates. And these are the coordinates of the vector.
This means that the vectors and are equal. The theorem is proven.
The theorem gives the following method for constructing the sum of arbitrary vectors and . It is necessary to set aside the vector from the end of the vector, equal to the vector. Then the vector, the beginning of which coincides with the beginning of the vector, and the end with the end of the vector, will be the sum of the vectors and. This method of obtaining the sum of two vectors is called<<правилом треугольника >> vector addition.
For vectors with a common origin, their sum is depicted by the diagonal of a parallelogram built on these vectors (<<правило параллелограмма >>).
Really , A . Means 
.
The difference of vectors is the vector that, when added to the vector, gives the vector: . From here we find the coordinates of the vector:
5.Multiplying a vector by a number
Let us produce the vector ) by the number called the vector ), i.e. ) ).
From the definition of the operation of multiplying a vector by a number it follows that for any vector and numbers ,
For any two vectors and and numbers
Theorem. The absolute value of the vector is . The direction of the vector at coincides with the direction of the vector if and is opposite to the direction of the vector if .
The absolute value of the vector is equal.
Some physical quantities, such as force, speed, acceleration, etc., are characterized not only by numerical value, but also by direction. In this regard, it is convenient to represent the indicated physical quantities by directed segments.
A directed segment is called a vector. To denote vectors we will use lowercase Latin letters. Sometimes a vector is denoted by indicating the ends of the segment representing the vector. For example, in Figure 232, a, a vector is shown. Point A is called the beginning, and point B is the end of the vector. When designating a vector using the ends of the segment representing it, the beginning of the vector is always put in the first place. An arrow or line is placed above the letter designation. For example, entry a reads: “Vector a.”
Two half-lines are called identically directed if they are combined by parallel translation, that is, there is a parallel translation that transfers one half-line to another.
If half-lines a and are equally directed and half-lines and c are equally directed, then half-lines a and c are equally directed.
In Figure 232, b, the half-lines are equally directed, also equally directed, which means the half-lines are equally directed.
Two half-lines are called oppositely directed if each of them has the same direction as the half-line complementary to the other.
Vectors are called identically directed if the half-lines are identically directed. Oppositely directed vectors are determined similarly.
The absolute value (or modulus) of a vector is the length of the segment representing the vector. The absolute value of vector a is denoted
Two vectors are said to be equal if they are combined by parallel translation. This means that there is a parallel translation that takes the start and end of one vector to the start and end of another vector, respectively.
Equal vectors have the same direction and are equal in absolute value.
Conversely: if the vectors are identically directed and equal in absolute value, then they are equal.
The beginning of a vector can coincide with its end. Such a vector is called the zero vector. The zero vector is denoted by a zero with a dash. They don't talk about the direction of the zero vector. The absolute value of the zero vector is considered equal to zero. All zero vectors are equal by definition.
A VECTOR Geometrically, vectors are represented by directed segments. A directed segment is called a vector. A vector is characterized by the following elements of direction, starting point(point of application), length (vector modulus). If the beginning of a vector is point A, its end is point B, then the vector is designated AB or a. C D B vectors: AB; SD A B
Absolute value. Equal vectors. Absolute value. Equal vectors. The absolute value (or modulus) of a vector is the length of the segment representing the vector. The absolute value of vector a is denoted by | a |. Two vectors are called equal if they are combined by parallel translation. vector ABCD is a parallelogram, AB = DC
Codirectional and oppositely directed vectors If vectors a and b are collinear and their rays are codirectional, then vectors a and b are called codirectional. Designated as a. If vectors a and d are collinear, and their rays are not codirectional, then vectors a and d are called oppositely directed. Designated a d. Zero vector It is agreed that the zero vector is codirectional with any vector.
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