Ryabushko A.P. IDZ 3.1 option 9

No. 1.9. Four points are given: A1(7;5;3); A2(9;4;4); A3(4;5;7); A4(7;9;6). It is necessary to create equations: a) plane A1A2A3; b) straight A1A2; c) straight line A4M, perpendicular to the plane A1A2A3; d) straight line A3N parallel to straight line A1A2; e) a plane passing through point A4 and perpendicular to straight line A1A2. It is also necessary to calculate: e) the sine of the angle between straight line A1A4 and plane A1A2A3; g) cosine of the angle between the coordinate plane Oxy and the plane A1A2A3.

a) To compile the equation of the plane A1A2A3, you can use the formula for the general equation of the plane: Ax + By + Cz + D = 0, where the coefficients A, B and C determine the direction coefficients of the normal to the plane. In order to find the guiding coefficients, it is necessary to take the vector product of two vectors lying on the plane, for example:

Vector A1A2: (9-7; 4-5; 4-3) = (2; -1; 1) Vector A1A3: (4-7; 5-5; 7-3) = (-3; 0; 4)

The direction coefficients can be found as the coordinates of the vector obtained as a result of the cross product: Normal to the plane: (-4; -10; -2)

Substituting one of the given points, for example, A1, into the equation of the plane, we obtain the value of the coefficient D: 7*(-4) + 5*(-10) + 3*(-2) + D = 0 D = 68

Thus, the equation of the plane A1A2A3 has the form: -4x - 10y - 2z + 68 = 0

b) To compile the equation of straight line A1A2, you can use the parametric form of the straight line equation: x = x1 + at y = y1 + bt z = z1 + ct

where (x1, y1, z1) are the coordinates of point A1, and (a, b, c) are the directing coefficients of the line.

The directing coefficients of a straight line can be found as the difference between the coordinates of the end and start points: a = 9 - 7 = 2 b = 4 - 5 = -1 c = 4 - 3 = 1

Thus, the equation of line A1A2 has the form: x = 7 + 2t y = 5 - t z = 3 + t

c) To compose the equation of the line A4M perpendicular to the plane A1A2A3, it is necessary to use the property that a vector drawn from the point of intersection of the line with the plane in the direction normal to the plane will lie on this line.

The normal to the plane A1A2A3 was found in point a) and is equal to (-4; -10; -2). Let's find the coordinates of the point of intersection of straight line A4M with plane A1A2A3; to do this, substitute the coordinates of point A4 into the equation of the plane:

-47 - 109 - 2*6 + D = 0 D = 128

Thus, the equation of the plane passing through point A4 is: -4x - 10y - 2z + 128 = 0

Let's find the directing coefficients of straight line A4M using the found normal to the plane A1A2A3: a = -4 b = -10 c = -2

Thus, the equation of line A4M is: x = 7 - 4t y = 9 - 10t z = 6 - 2t

d) To compose the equation of straight line A3N parallel to straight line A1A2, you can use the fact that the direction vectors of parallel straight lines are collinear. Let's find the direction vector of straight line A1A2:

(9-7; 4-5; 4-3) = (2; -1; 1)

Thus, the guiding coefficients of straight line A3N will also be equal to (2; -1; 1).

In order to find the equation of the line A3N, it is necessary to find the coordinates of a point lying on this line. To do this, select any of the given points, for example, A3, and substitute its coordinates into the equation of the parametric form of the straight line:

x = 4 + 2t y = 5 - t z = 7 + t

Thus, the equation of straight line A3N has the form: x = 4 + 2t y = 5 - t z = 7 + t

e) To construct the equation of a plane passing through point A4 and perpendicular to line A1A2, you can use the same property that a vector drawn from the point of intersection of the plane with the line in the direction normal to the plane will lie on this line.

The directing coefficients of straight line A1A2 were found in point b) and are equal to (2; -1; 1). Let's find the normal to this line using the property that the normal to the line is perpendicular to its direction vector:

(2; -1; 1) x (0; 0; 1) = (-1; -2; -2)

Thus, the direction coefficients of the plane perpendicular to straight line A1A2 are equal to (-1; -2; -2).

In order to find the equation of this plane, it is necessary to substitute the coordinates of point A4 into the equation of the general equation of the plane and find the value of the coefficient D:

-17 - 29 - 2*6 + D = 0 D = 31

Thus, the equation of the plane passing through point A4 and perpendicular

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Ryabushko A.P. IDZ 3.1 version 9 is a geometry problem in which you need to create equations of the plane, straight lines and calculate the values ​​of trigonometric functions for given points in space. The problem is given four points A1(7;5;3); A2(9;4;4); A3(4;5;7); A4(7;9;6), and required:

a) Write down the equation of the plane A1A2A3. b) Write down the equation of straight line A1A2. c) Create an equation for straight line A4M perpendicular to plane A1A2A3. d) Create an equation for straight line A3N parallel to straight line A1A2. e) Write an equation for a plane passing through point A4 and perpendicular to straight line A1A2. f) Calculate the sine of the angle between straight line A1A4 and plane A1A2A3. g) Calculate the cosine of the angle between the coordinate plane Oxy and the plane A1A2A3.

To solve the problem, knowledge of spatial geometry, vector algebra, trigonometry and algebra is required. Solving the problem can be useful for students and schoolchildren studying these topics as part of geometry and mathematics courses.

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Ryabushko A.P. IDZ 3.1 option 9 is a math task that consists of three numbers.

In the first issue, the coordinates of four points in three-dimensional space are given, and it is required to create equations for various geometric objects, such as a plane, a straight line, etc. You also need to find the values ​​of some trigonometric functions.

In the second issue, you need to create general equations for a straight line formed by the intersection of two planes.

In the third number you need to find the value of the parameter C at which the two given planes will be perpendicular.

If you have any questions, you can contact the seller listed in the seller information.

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