Product of a vector and a number
Goals: introduce the concept of multiplying a vector by a number; consider the basic properties of multiplying a vector by a number.
Lesson progress
I. Learning new material(lecture).
1. It is advisable to give an example at the beginning of the lecture that leads to the definition of the product of a vector and a number, in particular this:
The car moves 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, its 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. Determination of the product of a vector and a number, its designation: (Fig. 260).
3. Write down in your notebooks:
1) the product of any vector and the number zero is a zero vector;
2) for any number k and any vector, the vectors and are collinear.
4. Basic properties of multiplying a vector by a number:
For any numbers k, l and any vectors the equalities are valid:
1°. 
(combinative law) (Fig. 261);
2°. 
(first distributive law) (Fig. 262);
3°. 
(second distributive law) (Fig. 263).
Note. The properties of operations 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.
The work zero vector for any number a zero vector is considered. For any number k and any vector a, the vectors a and ka are collinear. From this definition it also follows that the product of any vector and the number zero is a zero vector.
Slide 38 from the presentation "Vectors" 11th grade. The size of the archive with the presentation is 614 KB.Geometry 11th grade
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Vector subtraction
Vector addition
Vectors can be added. The resulting vector is the sum of both vectors and determines 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 Lviv. How far will you be from your home while visiting your wife’s mother?
To answer this question, you need to draw a vector from the starting point of the trip (Kyiv) to the final point (Lviv). The new vector determines the result of the entire journey from beginning to end.
- Vector A - Kyiv-Moscow
- Vector B - Moscow-Lviv
- Vector C - Kyiv-Lviv
C = A+B, where C - vector sum 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. There is usually a dash at the top.
For example, a vector directed from the point A to the point B, can be designated a,
Zero vector 0 or 0 is a vector whose starting and ending 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 representing 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.
Vector addition. Since vectors are directed segments, then their addition can be performed geometrically.(Algebraic addition of vectors is described below, in paragraph “Unit orthogonal vectors"). Let's assume 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 starting point coincides with the end point of the second vector;
b) geometric addition , i.e. constructing the resulting vector coming from starting point fixed vector k end point transferred vector.
Subtraction of vectors. This operation is reduced to the previous one by replacing the subtrahend vector with its opposite one: a–b =a+ (– b) .
Laws of addition.
I.a+ b = b + a(Transitional law).
II. (a+ b) + c = a+ (b + c) (Combinative law).
III. a+ 0= a.
IV. a+ (-a) = 0 .
Laws for multiplying a vector by a number.
I. 1 · a= a,0 · a= 0 , m· 0 = 0 , ( – 1) · a= – a.
II. m a = a m,| m a| = | m | · | a | .
III. m (n a) = (m n) a .(C o m b e t a l
law of multiplication by number).
IV. (m+n) a= m a + n a,(DISTRIBUTIONAL
m(a+ b)= m a + m b . law of multiplication by number).
Dot product of vectors. __ __
Angle between non-zero vectors AB And CD- this is the angle formed by vectors with their parallel transfer until the points are combined A And C. Dot product of vectors a And b is called a number equal to the product of their lengths and the cosine of the angle between them:
If one of the vectors is zero, then their scalar product, in accordance with the definition, is equal to 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:
Dot product ( a , a), equal to | a| 2, called scalar square. Vector length a and its scalar square are related by the relation:
Dot product of two vectors:
- positively, if the angle between the vectors spicy;
- negative, if the angle between the vectors blunt.
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) . (Transitional law)
II. (m a , b) = m(a, b) .
III.(a+b,c) = (a , c) + (b,c). (Distributive law
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 are codirectional at and oppositely directed at. The product of a zero vector by any number is considered to be a zero vector. The product of a zero vector and a number is a vector whose length is equal, and the vectors and are codirectional at and oppositely directed at. The product of a zero vector by any number is considered to be a zero vector.
The product of a vector and a number is denoted as follows: The product of a vector and 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 and the number zero is a zero vector. The product of any vector and the number zero is a zero vector.
For any vectors, and any numbers, the equalities are valid: For any vectors, and any numbers, the equalities are valid: (combinative law) (combinative law) (first distributive law) (first distributive law) (second distributive law) (second distributive law)
(-1) is the vector opposite to the vector, i.e. (-1) =-. The lengths of the vectors (-1) and are equal to:. (-1) is the vector opposite to the vector, i.e. (-1) =-. The lengths of the vectors (-1) and are equal to:. 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 is a number such that. If the vectors and are collinear and, then there is a number such that.
Coplanar vectors Vectors are called 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 you put off a vector equal from point O. Vectors, and are coplanar, since if you put off a vector equal from point O, you get a vector, and vectors, you get a vector, and vectors, and lie in the same plane OSCE. Vectors, and are not coplanar, since the vector does not lie in the OAB plane. and lie in the same OCE plane. Vectors, and are not coplanar, since the vector does not lie in the OAB plane.
Proof of the property The vectors are not collinear (if the vectors are collinear, then the coplanarity of the vectors is obvious). Let's put it aside from arbitrary point O vectors and (fig.). Vectors and lie in the OAB plane. The vectors lie in the same plane. The vectors are not collinear (if the vectors are collinear, then the coplanarity of the vectors is obvious). Let us plot the vectors and from an arbitrary point O (Fig.). Vectors and lie in the OAB plane. In the same plane lie the vectors, and therefore their sum-vector, and therefore 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 coplanar. coplanar.
If the vectors and are coplanar, and the vectors and are not collinear, then the vector can be expanded into vectors. If the vectors and are coplanar, and the vectors and are not collinear, then the vector can be expanded into vectors and (i.e., represented in the form), and (t .i.e., represent in the form), and the expansion coefficients (i.e., numbers and in the formula) are determined in a unique way. Moreover, the expansion coefficients (i.e., the numbers and in the formula) are determined in a unique way.
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