Solution to problem 15.3.9 from the collection of Kepe O.E.

Problem 15.3.9 from the collection of Kepe O.?. is formulated as follows: “Find the angle between two planes passing through three points in space given by the coordinates.”

To solve the problem you need to perform the following steps:

1. Find equations of planes passing through given points.
2. Write down the equations of planes in general form.
3. Find the angle between the planes using the formula: cos(angle) = (a1a2 + b1b2 + c1*c2) / (sqrt(a1^2 + b1^2 + c1^2) * sqrt(a2^2 + b2^2 + c2^2)), where a1, b1, c1 and a2, b2, c2 - coefficients of plane equations.

Solution to problem 15.3.9 from the collection of Kepe O.?. may be of interest to students and professionals in the fields of geometry and linear algebra.

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Problem 15.3.9 from the collection of Kepe O.?. is as follows: given a matrix of size n x n, in which each element is equal to 0 or 1. It is necessary to determine whether it is possible to select a set of k rows of the matrix such that each column contains at least one unit.

To solve this problem, you can use the incidence matrix method and the algorithm for finding the maximum matching in a bipartite graph. First, an incidence matrix is ​​constructed in which the rows correspond to the rows of the matrix, and the columns correspond to the columns of the matrix, and the element of the incidence matrix is ​​equal to 1 if the corresponding matrix element is equal to 1, and 0 otherwise. Then a bipartite graph is constructed in which the vertices of the first part correspond to the rows of the matrix, the vertices of the second part correspond to the columns of the matrix, and an edge is drawn between the vertices if the corresponding element of the incidence matrix is ​​equal to 1.

After constructing the graph, a maximum matching search algorithm is applied, which allows you to find the maximum number of graph edges that do not have common vertices. If the number of edges in the found matching is equal to k, then the answer to the problem is positive, otherwise it is negative.

Solution to problem 15.3.9 from the collection of Kepe O.?. consists in determining the speed of ring D at point C if its speed at point A is zero. To do this, it is necessary to use the laws of conservation of energy and mechanics.

Wire ABC is a curved arc of a circle of radii r1 = 1 m, r2 = 2 m, located in a vertical plane. A ring D of mass m can slide without friction along a wire.

Using the law of conservation of energy, we can write that the kinetic energy of ring D at point A is zero, since its speed at this point is zero. Therefore, at point C the kinetic energy of ring D is equal to the potential energy of the ring at point A:

mgh = (1/2)mv^2,

where m is the mass of the ring, g is the acceleration of gravity, h is the height of the ring from point A to point C, v is the speed of the ring at point C.

The lifting height of the ring can be determined from geometric considerations:

h = r2 - r1 = 1 m.

Thus, substituting the known values ​​into the equation, we get:

mg = (1/2)mv^2, v^2 = 2gh = 2g(r2 - r1), v = sqrt(2g(r2 - r1)),

where sqrt is the square root, g ≈ 9.81 m/s² is the acceleration of gravity.

Substituting numerical values, we get:

v = sqrt(2 * 9.81 * (2 - 1)) ≈ 9.90 m/s.

Answer: the speed of ring D at point C is 9.90 m/s.

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