Ryabushko A.P. IDZ 2.2 option 3

IDZ - 2.2 No. 1.3. Vectors are given. It is necessary: ​​a) calculate the mixed product of three vectors; b) find the modulus of the vector product; c) calculate the scalar product of two vectors; d) check whether two vectors are collinear or orthogonal; e) check whether the three vectors are coplanar.

Given vectors: a(2;-4;-2); b(7;3;0); c(3;5;-7).

To calculate the mixed product of vectors a, b and c, it is necessary to use the formula to find the determinant of the matrix composed of the coordinates of these vectors: (a, b, c) = | 2 -4 -2 | | 7 3 0 | | 3 5 -7 | = (-94; -13; 59)

The modulus of the vector product of vectors a and b can be found by the formula: |a x b| = √(ax^2 + ay^2 + az^2) = √(9^2 + 14^2 + 29^2) = √986 ≈ 31.39

The scalar product of vectors a and b is calculated by the formula: a * b = 2*7 + (-4)*3 + (-2)*0 = 8

To determine the collinearity or orthogonality of vectors, it is necessary to calculate their scalar product. If it is equal to 0, then the vectors are orthogonal, if it is equal to the product of their lengths, then the vectors are collinear. Let's calculate the scalar product of vectors a and b: a * b = 8, not equal to 0 and not equal to the product of the lengths of the vectors, which means the vectors are not collinear and not orthogonal.

To determine the coplanarity of three vectors, it is necessary to check whether they lie in the same plane. To do this, you can check whether the mixed product of vectors a, b and c is equal to zero: (a, b, c) = (-94; -13; 59), not equal to 0, which means the vectors are not coplanar.

No. 2.3. The vertices of the pyramid are located at points A(1;3;1); B(–1;4;6); C(–2;–3;4); D(3;4;–4).

To solve the problem, it is necessary to find the volume of the pyramid, which can be calculated using the formula: V = (1/3) * S * h, where S is the area of ​​the base of the pyramid, and h is the height of the pyramid.

The area of ​​the base of the pyramid can be found as the area of ​​the parallelogram formed by the vectors AB and AC: S = |AB x AC| = |(-2;-1;5)| = √30

The height of the pyramid can be found as the distance from the vertex D to the plane containing the base ABC. To do this, you need to find the equation of the plane passing through points A, B and C, and substitute the coordinates of vertex D into this equation. The plane equation can be found as the product of the vectors AB and AC: n = AB x AC = (-2;-1;5) Plane equation: -2x- y + 5z = 0

Now you can find the distance from point D to the plane using the formula: h = |(AD * n) / |n||, where AD is the vector connecting vertex D to any point on the plane, and |n| - vector length n.

Let's take point A on the plane and find the vector AD: AD = D - A = (3;1;-5)

Let's find the length of vector n: |n| = √(4^2 + 1^2 + 5^2) = √42

Now you can calculate the height of the pyramid: h = |(AD * n) / |n|| = |(-31) / √42| ≈ 4.81

In total, the volume of the pyramid is: V = (1/3) * S * h = (1/3) * √30 * 4.81 ≈ 2.07

No. 3.3. Force F(2;19;–4) is applied to point A(5;3;4). Calculate: a) the work of force in the case when the point of its application, moving rectilinearly, moves to point B(6;–4;–1); b) modulus of the moment of force relative to point B.

To solve the problem, it is necessary to find the work done by the force and the modulus of the moment of this force relative to point B.

a) The work of force F when moving the application point from point A to point B is calculated by the formula: A = F * Δr, where Δr is the displacement vector of the application point.

Let's find the displacement vector Δr by subtracting the coordinates of point A from the coordinates of point B: Δr = B - A = (1; -7; -5)

Now you can calculate the work of force: A = F * Δr = (2; 19; -4) * (1; -7; -5) = -205

Answer: the work done by force F when moving the point of application from point A to point B is equal to -205.

b) The moment of force is calculated by the formula: M = r x F, where r is the vector from the point of application of the force to the point around which the moment is calculated.

Let's find the vector r from point B to the point of application of force F: r = A - B = (-1; 7; 5)

Now you can calculate the moment of force: M = r x F = (-1; 7; 5) x (2; 19; -4) = (-93; 18; -33)

The modulus of the moment of force is equal to the length of this vector: |M| = √((-93)^2 + 18^2 + (-33)^2) ≈ 98.69

Answer: the modulus of the moment of force F relative to point B is approximately 98.69.

"Ryabushko A.P. IDZ 2.2 option 3" is a digital product that is a set of problems in linear algebra. You can purchase this product from the digital goods store.

Each task is colorfully designed in HTML markup, which makes using this product more convenient and enjoyable. You can easily find the problem you need, read the conditions and get a solution.

This digital product is ideal for students studying linear algebra, as well as for anyone who wants to improve their knowledge in this field. By purchasing "Ryabushko A.P. IDZ 2.2 option 3", you get a convenient tool for independent preparation and practice of the material.

"Ryabushko A.P. IDZ 2.2 option 3" is an electronic file with a description of problems in vector algebra. The file contains three tasks in which it is necessary to calculate the mixed product of three vectors, find the modulus of the vector product, calculate the scalar product of two vectors, check whether two vectors are collinear or orthogonal, check whether three vectors are coplanar, find the volume of the pyramid, work of force and the modulus of the moment of force relative to the point. The product is presented as an electronic document in PDF or DOCX format and can be downloaded after payment.

***

Ryabushko A.P. IDZ 2.2 option 3 is a task for doing homework in linear algebra. The task consists of three numbers, each of which contains several subtasks.

The first number contains three vectors a(2;-4;-2), b(7;3;0) and c(3;5;-7). You need to calculate the mixed product of three vectors, find the modulus of the cross product, calculate the dot product of two vectors, check whether two vectors are collinear or orthogonal, and check whether three vectors are coplanar.

The second issue gives the coordinates of the vertices of the pyramid: A(1;3;1), B(–1;4;6), C(–2;–3;4), D(3;4;–4). The task requires calculating some parameters of the pyramid, but specific subtasks are not specified.

The third issue gives the coordinates of points A(5;3;4) and B(6;–4;–1), as well as the force F(2;19;–4) applied to point A. It is necessary to calculate the work of the force in the case , when the point of its application moves rectilinearly and moves to point B(6;–4;–1), as well as the modulus of the moment of force relative to point B.

If you have any questions about completing the task, you can contact the seller at the email address provided in the seller information.

***

1. A digital product is a convenient way to obtain information or entertainment without having to purchase physical copies.
2. A digital product can be easily downloaded and used on various devices such as computers, tablets and smartphones.
3. The digital product can be delivered immediately, without having to wait for delivery.
4. A digital product can be updated and improved without the need to purchase a new physical copy.
5. A digital product can be accessed from anywhere in the world where there is an Internet connection.
6. Digital goods can be easily stored and stored on devices without the need for physical storage space.
7. A digital product can be a more environmentally friendly option because it does not require the use of paper and other materials to create physical copies.

Peculiarities:

Digital goods can be delivered quickly and easily without having to wait for delivery.

Digital goods are often available at a lower price than physical goods.

Digital goods can be easily stored and organized on a computer or in the cloud, making them easier to access and more convenient to use.

Digital goods may be