Option 18 IDZ 2.2

No. 1.18. Problem condition: given vectors a(9;-3;1), b(3;-15;21), c(1;-5;7). Necessary:

a) Calculate the mixed product of three vectors. The mixed product of vectors a, b and c is equal to their mixed component and is determined by the formula: (a, b, c) = a·(b x c), where x is the sign of the vector product, · is the sign of the scalar product. We calculate the vector product b x c: b x c = (15·7 -21·(-5); 21·1 - 3·7; 3·(-5) - 15·1) = (162; -132; -12) Now we find scalar product of a and the resulting vector: a·(b x c) = 9·162 + (-3)·(-132) + 1·(-12) = 1575

b) Find the modulus of the vector product. The magnitude of the vector product b x c is equal to the length of the vector equal to this product. Calculate the length: |b x c| = √(162² + (-132)² + (-12)²) ≈ 214.97

c) Calculate the scalar product of two vectors. The scalar product of two vectors a and b is equal to the sum of the products of their corresponding coordinates: a·b = 9·3 + (-3)·(-15) + 1·21 = 81

d) Check whether two vectors are collinear or orthogonal. Two nonzero vectors are collinear if one is a multiple of the other. Two non-zero vectors are orthogonal if their scalar product is 0. Check for vectors a and b: a/b = (9/3; (-3)/(-15); 1/21) = (3; 0.2; 0.05) a·b = 81 The vectors are neither collinear nor orthogonal.

e) Check whether the three vectors are coplanar. Three vectors are coplanar if they lie in the same plane. One possible check is to calculate the mixed product of the vectors and check whether it is zero: (a, b, c) = a·(b x c) = 1575 ≠ 0 Since the mixed product is not zero, the vectors are not coplanar.

No. 2.18. Problem condition: the vertices of the pyramid are located at points A(5;-4;4), B(-4;-6;5), C(3;2;-7), D(6;2;-9).

To solve the problem, you need to find the volume of the pyramid using the coordinates of the vertices. The volume of a pyramid with its vertex at point V is equal to one-sixth of the volume of a parallelepiped, one of whose faces is parallel to the coordinate axis and passes through point V, and the other two faces pass through the adjacent vertices of the pyramid.

We calculate the coordinates of the vectors leaving vertex A: AB = (-9; -2; 1), AC = (-2; 6; -11), AD = (1; 6;-13) Now we find the mixed product of the vectors AB, AC and AD to find the volume of the parallelepiped: V_par = (AB, AC, AD) = AB · (AC x AD) = (-9; -2; 1) · (-76; -16; 54) = 2142 Volume of the pyramid with the vertex at point A will be equal to one sixth of the volume of the found parallelepiped: V_pir = V_par / 6 ≈ 357

No. 3.18. Problem condition: given are three forces P(-5;8;4), Q(6;-7;3), R(3;1;-5) applied to point A(2;-4;7). Necessary:

a) Calculate the work produced by the resultant of these forces when the point of its application, moving rectilinearly, moves to point B(0;7;4). The work done by the resultant force is equal to the scalar product of the resultant and the displacement of the point of its application: A_B = (0-2; 7-(-4); 4-7) = (-2; 11; -3) P + Q + R = (-5+6+3; 8-7+1; 4+3-5) = (4; 2; 2) W = (P + Q + R) · A_B = (-8; 22; -6) · (4; 2; 2) = 8

b) Calculate the magnitude of the moment of the resultant of these forces relative to point B. The moment of force relative to the point is equal to the vector product of the radius vector of the point relative to the center of the moment by the resultant of forces: B_A = -A_B = (2; -11; 3) M_B = B_A x (P + Q + R) = (2; -11; 3) x (4; 2; 2) = (-26; -4; -50) The magnitude of the moment is equal to the length of this vector: |M_B| = √((-26)² + (-4)² + (-50)²) ≈ 51.42

This product, called "Option 18 IDZ 2.2" is a digital product that is intended for educational use. This product is a set of problems to solve, including problems in mathematics and physics.

The product design is made in a beautiful html format, which ensures ease of use and improves the overall perception of information. Each task is designed as a separate block, which allows you to quickly navigate and find the information you need.

"Option 18 IDZ 2.2" is an excellent choice for students and students who want to improve their knowledge in mathematics and physics. This digital product provides an opportunity to effectively train and improve your knowledge in these areas, which can be very beneficial for your academic and professional career.

"Option 18 IDZ 2.2" is a set of tasks in mathematics and physics, including three tasks.

In problem No. 1.18, three vectors a, b and c are given, and several problems must be solved: calculate the mixed product of three vectors (a, b, c); find the modulus of the vector product of vectors b and c; calculate the scalar product of vectors a and b; check whether two vectors a and b are collinear or orthogonal; check whether three vectors a, b and c are coplanar.

In problem No. 2.18, the coordinates of the vertices of the pyramid are given, and you need to find the volume of the pyramid. To do this, you need to calculate the coordinates of the vectors emanating from vertex A, then find the mixed product of the vectors AB, AC and AD to find the volume of the parallelepiped. The volume of the pyramid with its vertex at point A will be equal to one-sixth of the volume of the found parallelepiped.

Problem No. 3.18 is given three forces P, Q and R applied to point A, and it is required to calculate the work done by the resultant of these forces when the point of its application moves to point B, as well as the magnitude of the moment of the resultant of these forces relative to point B. To calculate work, it is necessary to find the resultant of the forces and the displacement of the point of its application, and then calculate the scalar product of these vectors. To calculate the moment, you need to find the radius vector of point B relative to point B, and then calculate the vector product of this radius vector and the resultant force.

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From the provided product description (option 18 IDZ 2.2) we can conclude that this is a problem book on linear algebra and mechanics. It contains three tasks.

The first task consists of five subtasks and is associated with the calculation of various characteristics of vectors (mixed product, modulus of the vector product, scalar product) and determination of their properties (collinearity, orthogonality, coplanarity). To solve the problem, three vectors a(9;-3;1), b(3;-15;21) and c(1;-5;7) are given.

The second problem involves calculating the volume of a pyramid given by the coordinates of its vertices A(5;-4;4), B(-4;-6;5), C(3;2;-7) and D(6;2;- 9).

The third task involves calculating the work and moment of forces applied to point A(2;-4;7) and moved to point B(0;7;4). To solve the problem, three forces are given: P(-5;8;4), Q(6;-7;3) and R(3;1;-5).

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