Option 20 IDZ 3.1

Download Mirror

No. 1.20. The problem gives the coordinates of four points in three-dimensional space: A1(1;–1;3); A2(6;5;8); A3(3;5;8); A4(8;4;1). The following tasks need to be solved:

a) Find the equation of the plane passing through points A1, A2 and A3. To do this, you can use the formula for the general equation of the plane: Ax + By + Cz + D = 0, where A, B and C are the coefficients determined by the vector product of two vectors lying in the plane, and D is the free term determined by substituting the coordinates of one of the points . The resulting equation will look like: 4x + 13y - 11z - 33 = 0.

b) Find the equation of the line passing through points A1 and A2. To do this, you can use the formula for the parametric equation of a straight line: x = x1 + at, y = y1 + bt, z = z1 + ct, where a, b and c are guiding coefficients, defined as the difference between the corresponding coordinates of points, and t is a parameter. The resulting equation will look like: x = 1 + 5t, y = -1 + 6t, z = 3 + 5t.

c) Find the equation of the line passing through points A4 and M and perpendicular to the plane A1A2A3. To do this, you can use the formula for the equation of a straight line in segment-parametric form: x = x1 + (x2 - x1)t, y = y1 + (y2 - y1)t, z = z1 + (z2 - z1)t, where x1, y1 , z1 are the coordinates of point A4, x2, y2, z2 are the coordinates of point M, and t is a parameter. To determine the direction vector of the straight line, it is necessary to take the vector product of the vectors MA4 and the normal to the plane A1A2A3. The resulting equation will look like: x = 8 - 5t, y = 4 - 9t, z = 1 + 7t.

d) Find the equation of a line passing through points A3 and N and parallel to line A1A2. To do this, you can use the formula for the parametric equation of a straight line, similar to the equation of straight line A1A2: x = 3 + t, y = 5 + 2t, z = 8 + 3t.

e) Find the equation of the plane passing through point A4 and perpendicular to straight line A1A2. To do this, you can use the formula for the general equation of the plane, similar to the equation of the plane A1A2A3, but with different coefficients. The direction vector of the plane will coincide with the direction vector of straight line A1A2. The resulting equation will look like: 6x - 5y - 7z + 46 = 0.

f) Find the sine of the angle between straight line A1A4 and plane A1A2A3. To do this, you need to find the scalar product of the vectors corresponding to the directions of these lines, and then divide the resulting value by the product of the absolute values of these vectors. The sine of the angle between them will be equal to the modulus of this product divided by the product of the moduli of the vectors. The resulting value will be 0.82.

g) Find the cosine of the angle between the coordinate plane Oxy and the plane A1A2A3. To do this, you need to find the scalar product of vectors normal to these planes, and then divide the resulting value by the product of the moduli of these vectors. The cosine of the angle between them will be equal to the resulting value. The resulting value will be 0.39.

No. 2.20. To compile the equation of a plane passing through the Oy axis and the point M(3;–5;2), it is necessary to use the formula for the general equation of the plane: Ax + By + Cz + D = 0. Since the plane passes through the Oy axis, then the coefficient A and C will be equal to 0. To determine the coefficient B, it is necessary to substitute the coordinates of point M into the equation and solve the equation with respect to B. The resulting equation will look like: 5y + D = 0. To determine the free term D, it is necessary to substitute the coordinates of point M into the equation and solve the equation with respect to D. The resulting equation will look like: D = -25. Thus, the equation of the plane will be: 5y - 25 = 0.

No. 3.20. In order to find the value of D at which the straight line intersects the Oz axis, it is necessary to create an equation of the straight line in segment-parametric form: x = x1 + (x2 - x1)t, y = y1 + (y2 - y1)t, z = z1 + (z2 - z1)t, where x1, y1, z1 are the coordinates of the point through which the line passes, x2, y2, z2 are the coordinates of another point on the line, and t is a parameter. Then you need to substitute the coordinates of the straight line into the equation of the Oz axis, which has the form z = 0, and solve the equation with respect to the parameter t. The resulting t value will allow you to find the z coordinate of the point of intersection of the line with the Oz axis.

The product "Option 20 IDZ 3.1" is a digital product intended for use for educational purposes. It is available in the digital store and is a set of math problems.

Each problem includes a set of data that must be processed and solved using appropriate mathematical methods. All tasks are completed in accordance with the requirements of the curriculum and can be used both for independent study and for preparing for exams.

The product design is made in a beautiful html format, which ensures ease of use and a pleasant visual experience. Each task is presented in a separate block, which makes it easy to navigate the material and quickly find the necessary data.

"Option 20 IDZ 3.1" is an excellent choice for students and anyone who is interested in mathematics and wants to improve their knowledge in this area. Thanks to its convenient design and accessibility, this product will become a reliable assistant in studying and preparing for exams.

this is mathematical task No. 1.20, in which the coordinates of four points in three-dimensional space are given, and it is also necessary to solve several problems related to determining the equations of planes and lines passing through these points. The problems use formulas for general and parametric equations of planes and lines, as well as vector and scalar products of vectors. For example, it is necessary to find the equations of planes passing through certain points and given lines, as well as find the angles between these lines. Solving problems will help improve your understanding of 3D geometry and strengthen your math skills.

***

Option 20 IDZ 3.1 is a geometry task that consists of three parts.

Part No. 1.20. Given four points A1(1;–1;3); A2(6;5;8); A3(3;5;8); A4(8;4;1). Necessary:

a) draw up an equation of the plane passing through points A1, A2 and A3;

b) compose an equation of a straight line passing through points A1 and A2;

c) create an equation for a straight line passing through point A4 and perpendicular to the plane passing through points A1, A2 and A3;

d) create an equation of a line parallel to the line passing through points A1 and A2, and passing through point A3;

e) draw up an equation of a plane passing through point A4 and perpendicular to the line passing through points A1 and A2;

f) calculate the sine of the angle between the straight line passing through points A1 and A4 and the plane passing through points A1, A2 and A3;

g) calculate the cosine of the angle between the coordinate plane Oxy and the plane passing through points A1, A2 and A3.

No. 2.20. It is necessary to create an equation for a plane passing through the Oy axis and the point M(3;–5;2).

No. 3.20. It is necessary to find the value of the parameter D in the equation of the straight line so that it intersects the Oz axis.

***

A very convenient and easy to use digital product.
Quickly gained access to the product, without having to wait for delivery.
The quality of the digital product exceeded my expectations.
I really liked that you could immediately start using the product without wasting time on installation.
Excellent value for money.
It is very convenient that you can use the product on several devices.
An excellent choice for those who want to save time and get a quality product.
The digital product met all my expectations and more.
Very simple and intuitive product interface.
A fast and efficient way to get the product you need anytime, anywhere.