IDZ Ryabushko 3.1 Option 14

№1

Given four points A1(3;5;4); A2(8;7;4); A3(5;10;4); A4(4;7;8). Make up equations:

1. Planes A1A2A3;
2. Direct A1A2;
3. Straight line A4M, perpendicular to plane A1A2A3;
4. Line A3N parallel to line A1A2;
5. A plane passing through point A4, perpendicular to straight line A1 A2;
6. Sine of the angle between straight line A1A4 and plane A1A2A3;
7. Cosine of the angle between the coordinate plane Oxy and the plane A1A2A3.

Answer:

1. Vectors connecting points A1, A2 and A3:

   A1A2(5;2;0); A2A3(-3;3;0); A3A1(-2; -5;0).

   Vector product of vectors A1A2 and A2A3:

   n(6;15;15).

   Equation of a plane passing through points A1, A2 and A3:

   6x + 15y + 15z - 105 = 0.

2. Vector connecting points A1 and A2:

   A1A2(5;2;0).

   Equation of a straight line passing through points A1 and A2:

   x = 3 + 5t, y = 5 + 2t, z = 4.

3. Vector connecting points A4 and M:

   A4M(-2;12;-5).

   Normal vector of the plane A1A2A3:

   n(6;15;15).

   Equation of a straight line passing through point A4 and perpendicular to the plane A1A2A3:

   x = 4 - 2t, y = 7 + 3t, z = 8 - t.

4. Vector parallel to straight line A1A2:

   A1A2(5;2;0).

   Equation of a line passing through point A3 and parallel to line A1A2:

   x = 5t, y = 10 + 2t, z = 4.

5. Vector parallel to straight line A1A2:

   A1A2(5;2;0).

   Equation of a plane passing through point A4 and perpendicular to line A1A2:

   5x + 2y - 15z + 34 = 0.

6. Vector connecting points A1 and A4:

   A1A4(1;2;-4).

   Normal vector of the plane A1A2A3:

   n(6;15;15).

   Sine of the angle between straight line A1A4 and plane A1A2A3:

   sinα(A1A4, n) / (|A1A4| * |n|) = (6 - 30 + 60) / (√21 * √450) ≈ 0.413.

7. Normal vector of the plane A1A2A3:

   n(6;15;15).

   Cosine of the angle between the coordinate plane Oxy and the plane A1A2A3:

   cosα = (n, k) / (|n| * |k|) = (6 + 0 + 0) / (√450) ≈ 0.267.

№2

Write an equation for a plane passing through points A(3;–1;2) and B(2;1;4) parallel to the vector a = (5;–2;–1).

Answer:

Vector AB(-1;2;2).

The normal vector of the desired plane must be parallel to vector a, therefore:

n(5;-2;-1).

Plane equation:

5(x - 3) - 2(y + 1) - (z - 2) = 0.

№3

Write an equation for a line passing through the point M(2;–5;3), parallel to the line 2x – y + 3z – 1 = 0 and 5x + 4y – z – 7 = 0.

Answer:

A vector parallel to the straight line 2x – y + 3z – 1 = 0 has coordinates:

a1(2;3;1).

A vector parallel to the straight line 5x + 4y – z – 7 = 0 has coordinates:

a2(4;-5;17).

Vector product of vectors a1 and a2:

n(53;33;-22).

Equation of a straight line:

x = 2 + 53t, y = -5 + 33t, z = 3 - 22t.

IDZ Ryabushko 3.1 Option 14

IDZ Ryabushko 3.1 Option 14 is a digital product intended for secondary school students and students studying mathematics. This product contains a selection of problems and exercises in various areas of mathematics, including algebra, geometry, trigonometry and calculus.

IDZ Ryabushko 3.1 Option 14 will help students consolidate and deepen their knowledge in mathematics, as well as prepare for exams and olympiads. It presents tasks of varying complexity, from simple exercises to more complex problems that will help develop logical thinking and the ability to apply mathematical methods in solving problems.

IDZ Ryabushko 3.1 Option 14 is presented in the form of an electronic document that can be downloaded immediately after payment. This is convenient for users, since they can start working with the product immediately after purchase. Beautiful html design of the product creates a pleasant visual impression and simplifies document navigation.

IDZ Ryabushko 3.1 Option 14 is an excellent choice for those who want to improve their knowledge in mathematics and get good results in exams and olympiads.

Product description: IDZ Ryabushko 3.1 Option 14 is a digital product intended for secondary school students and students studying mathematics. The product contains a selection of problems and exercises in various areas of mathematics, including algebra, geometry, trigonometry and calculus.

IDZ Ryabushko 3.1 Option 14 will help students consolidate and deepen their knowledge in mathematics, as well as prepare for exams and olympiads. It presents tasks of varying complexity, from simple exercises to more complex problems that will help develop logical thinking and the ability to apply mathematical methods in solving problems.

IDZ Ryabushko 3.1 Option 14 is presented in the form of an electronic document that can be downloaded immediately after payment. This is convenient for users, as they can start working with the product immediately after purchase. Beautiful product design creates a pleasant visual impression and simplifies document navigation.

Ryabushko's IDZ 3.1 Option 14 includes tasks that will help students develop problem-solving skills and learn to apply mathematical methods in practical situations. The product also contains solutions to problems, allowing students to check their answers and correct mistakes.

This product is an excellent choice for those who want to improve their knowledge in mathematics and get good results in exams and competitions.

***

IDZ Ryabushko 3.1 Option 14 is a mathematics task in which several geometric problems are given. The first task requires you to create equations for a plane and straight lines, as well as calculate some angles. In the second task, you need to create an equation of a plane passing through two given points and parallel to a given vector. In the third task, you need to create an equation of a line passing through a given point and parallel to two given lines.

***

1. An excellent digital product for preparing for the IPD. Solutions with detailed explanations help you quickly understand the material.
2. Thanks to the creators for such a convenient and functional product. IDZ Ryabushko 3.1 Option 14 helps me solve problems easily and quickly.
3. I am very pleased with this digital product. It helps me not only prepare for IPD, but also improve my understanding of the material.
4. IDZ Ryabushko 3.1 Option 14 is an excellent choice for those who want to get high marks in exams.
5. Many thanks to the creators for such a useful and convenient digital product. It helps me understand and remember the material better.
6. I recommend Ryabushko IDZ 3.1 Option 14 to anyone who wants to quickly and effectively prepare for exams.
7. With the help of this digital product, I can easily solve problems and get high scores in IPD. Highly recommend!.
8. IDZ Ryabushko 3.1 Option 14 is an indispensable assistant for everyone who wants to successfully cope with mathematics tasks.
9. Thank you very much for such a convenient and understandable digital product. With its help, I prepared for exams quickly and easily.
10. I am very pleased with Ryabushko IDZ 3.1 Option 14. This digital product really helps me understand the material and solve problems.

Peculiarities:

I am very pleased with the purchase of the IDS Ryabushko 3.1 Option 14 is a great digital product that helped me prepare for the exam.

IDZ Ryabushko 3.1 Option 14 is exactly what you need for those who want to pass the exam successfully and get a high mark.

Thanks to Ryabushko 3.1 Option 14, I was able to quickly and easily master the material and successfully pass the exam.

I really liked the Ryabushko 3.1 IDZ Option 14 is a convenient and practical digital product that helps me improve my level of knowledge.

I recommend Ryabushko IDZ 3.1 Option 14 to anyone who is serious about their studies - this is an excellent digital product that will help you pass the exam successfully.

With the Ryabushko ID 3.1 Option 14, I feel more confident in the exam - this is a great digital product that helps me understand the material better.

IDZ Ryabushko 3.1 Option 14 is an excellent choice for those who want to quickly and efficiently prepare for the exam and get a high mark.