Multiplication of a vector by a number The product of a zero vector by a number is a vector whose length is equal, and the vectors and are co-directed at and oppositely directed at. The product of a zero vector by any number is a zero vector. The product of a zero vector by a number is a vector whose length is equal, and the vectors and are co-directed at and oppositely directed at. The product of a zero vector by any number is a zero vector.

The product of a vector by a number is denoted as follows: The product of a vector by a number is denoted as follows: For any number and any vector, the vectors and are collinear. For any number and any vector, the vectors and are collinear. The product of any vector by zero is a zero vector. The product of any vector by zero is a zero vector.

For any vectors, and any numbers, equalities are true: For any vectors, and any numbers, equalities are true:

(-1) is a vector, opposite vector, i.e. (-1) =-. The lengths of the vectors (-1) and are:. (-1) is the vector opposite to the vector, i.e. (-1) =-. The lengths of the vectors (-1) and are:. If the vector is non-zero, then the vectors (-1) and are oppositely directed. If the vector is non-zero, then the vectors (-1) and are oppositely directed. IN PLANIMETRY IN PLANIMETRY If the vectors and are collinear and, then there exists a number such that. If the vectors and are collinear and, then there exists a number such that.

Coplanar Vectors Vectors are said to be coplanar if, when plotted from the same point, they lie in the same plane. Vectors are called coplanar if, when plotted from the same point, they lie in the same plane.

The figure shows a parallelepiped. The figure shows a parallelepiped. Vectors, and are coplanar, since if we set aside a vector equal to the point O Vectors, and are coplanar, since if we set aside a vector equal to the point O, then we get a vector, and vectors, we get a vector, and vectors, and lie in the same plane OSE. The vectors, and are not coplanar, since the vector does not lie in the OAB plane. and lie in the same OSE plane. The vectors, and are not coplanar, since the vector does not lie in the OAB plane.

Proof of the feature Vectors and are not collinear (if vectors and are collinear, then the complanarity of vectors and is obvious). Set aside from arbitrary point O vectors and (Fig.). The vectors and lie in the OAB plane. The vectors lie in the same plane. The vectors and are not collinear (if the vectors and are collinear, then the complanarity of the vectors and is obvious). Let us set aside the vectors and from an arbitrary point O (Fig.). The vectors and lie in the OAB plane. Vectors lie in the same plane, and hence their sum-vector, and hence their sum-vector, equal to vector. Vectors equal to vector. The vectors lie in the same plane, i.e. vectors, and lie in the same plane, i.e. vectors, and are coplanar. coplanar.

If the vectors, and are coplanar, and the vectors and are not collinear, then the vector can be decomposed into vectors. .i.e. represent in the form), and the expansion coefficients (i.e., the numbers and in the formula) are uniquely determined. moreover, the expansion coefficients (i.e., the numbers and in the formula) are uniquely determined.

The product of a zero vector by any number is a zero vector. For any number k and any vector a, the vectors a and ka are collinear. It also follows from this definition that the product of any vector by zero is a zero vector.

Slide 38 from the presentation "Vectors" Grade 11". The size of the archive with the presentation is 614 KB.

Geometry Grade 11

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Vector subtraction

Vector addition

Vectors can be added. The resulting vector is the sum of both vectors and defines the distance and direction. For example, you live in Kyiv and decided to visit old friends in Moscow, and from there make a visit to your beloved mother-in-law in Lvov. How far will you be from your home, visiting your wife's mother?

To answer this question, you need to draw a vector from starting point travel (Kyiv) and to the final (Lviv). The new vector determines the outcome of the entire journey from start to finish.

• Vector A - Kyiv-Moscow
• Vector B - Moscow-Lviv
• Vector C - Kyiv-Lviv

C \u003d A + B, where C - sum of vectors or the resulting vector

Top of page

Vectors can not only be added, but also subtracted! To do this, you need to combine the bases of the subtrahend and subtracting vectors and connect their ends with arrows:

• Vector A = C-B
• Vector B = C-A

23question:

A vector is a directed segment connecting two points in space or in a plane. Vectors are usually denoted either by small letters or by starting and ending points. Above is usually a dash.

For example, a vector directed from a point A to the point B, can be denoted a,

Zero vector 0 or 0 is a vector whose start and end points are the same, i.e. A=B.From here, 0 = – 0.

The length (modulus) of the vector a is the length of the segment AB that displays it, denoted by | a |. In particular, | | 0 | = 0.

The vectors are called collinear if their directed segments lie on parallel lines. Collinear vectors a and b are designated a|| b.

Three or more vectors are called coplanar if they lie in the same plane.

Addition of vectors. Since vectors are directed segments, then their addition can be performed geometrically.(Algebraic addition of vectors is described below, in the paragraph "Unit orthogonal vectors"). Let's pretend that

a=AB and b = CD,

then the vector __ __

a+ b = AB+ CD

is the result of two operations:

a)parallel transfer one of the vectors so that its start point coincides with the end point of the second vector;

b)geometric addition, i.e. constructing the resulting vector going from the starting point of the fixed vector to the end point of the transferred vector.

Subtraction of vectors. This operation is reduced to the previous one by replacing the subtracted vector with the opposite one: a-b =a+ (– b) .

The laws of addition.

I. a+ b = b + a(V erable law).

II. (a+ b) + c = a+ (b + c) (Combined law).

III. a+ 0= a.

IV. a+ (– a) = 0 .

Laws of multiplication of a vector by a number.

I. one · a= a,0 · a= 0 , m 0 = 0 , ( – one) · a= – a.

II. m a = a m,| m a| = | m | · | a | .

III. m (n a) = (m n) a .(Combined

law of multiplication).

IV. (m+n) a= m a + n a ,(Distributor

m(a+ b)= m a + m b . law of multiplication).

Scalar product of vectors. __ __

Angle between non-zero vectors AB and CD is the angle formed by vectors with them parallel transfer before matching points A and C. The scalar product of vectors a and b called a number equal to the product of their lengths by the cosine of the angle between them:

If one of the vectors is zero, then their scalar product, in accordance with the definition, is zero:

(a , 0) = (0,b) = 0 .

If both vectors are non-zero, then the cosine of the angle between them is calculated by the formula:

Scalar product ( a , a) equal to | a| 2, called scalar square. Vector length a and its scalar square are related by:

Dot product of two vectors:

- positively if the angle between the vectors spicy;

- negative if the angle between the vectors stupid.

The scalar product of two non-zero vectors is equal to zero if and only if the angle between them is right, i.e. when these vectors are perpendicular (orthogonal):

Properties of the scalar product. For any vectors a, b, c and any number m the following relations are valid:

I. (a , b) = (b, a) . (V erable law)

II. (m a , b) = m(a , b) .

III.(a + b , c) = (a , c) + (b, c). (Distributive law

The product of a vector by a number

Goals: introduce the concept of multiplication of a vector by a number; consider the basic properties of multiplication of a vector by a number.

During the classes

I. Learning new material(lecture).

1. It is advisable at the beginning of the lecture to give an example leading to the definition of the product of a vector by a number, in particular, this:

The car is moving in a straight line with a speed of . He is overtaken by a second car moving at twice the speed. A third car is moving towards them, whose speed is the same as that of the second car. How to express the speeds of the second and third cars in terms of the speed of the first car and how to represent these speeds using vectors?

2. Definition of the product of a vector by a number, its designation: (Fig. 260).

3. Write in notebooks:

1) the product of any vector by the number zero is a zero vector;

2) for any number k and any vector, the vectors and are collinear.

4. Basic properties of multiplication of a vector by a number:

For any numbers k, l and any vectors, the equalities are true:

1°. (associative law) (Fig. 261);

2°. (first distributive law) (Fig. 262);

3°. (second distributive law) (Fig. 263).

Note. The properties of actions on vectors that we have considered allow us to perform transformations in expressions containing sums, differences of vectors and products of vectors by numbers according to the same rules as in numerical expressions.

"It's called a vector" - Vectors. Addition of vectors Parallelogram rule. The second concept of a vector. Vector equality. Oppositely directed vectors. Building: Collinear vectors having opposite direction, are called oppositely directed vectors. Subtraction of vectors. Collinear vectors. End of vector.

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