IDZ Ryabushko 3.1 Option 3

1. Given four points A1(3;5;4); A2(5;8;3); A3(1;2;–2); A4(–1;0;2). It is necessary to create equations:
   1. Planes A1A2A3;
   2. Direct A1A2;
   3. Straight line A4M, perpendicular to plane A1A2A3;
   4. Line A3N parallel to line A1A2;
   5. The plane passing through point A4, perpendicular to straight line A1A2;
   You also need to calculate:
   1. Sine of the angle between straight line A1A4 and plane A1A2A3;
   2. Cosine of the angle between the coordinate plane Oxy and the plane A1A2A3.
2. It is necessary to find the distance from a point to a plane.
3. It is necessary to create an equation for a straight line passing through the point M(1;–3;3) and forming angles with the coordinate axes, respectively, of 60; 45 and 120.

   1. In order to find the equation of the plane A1A2A3, it is necessary to find the vector product of its two direction vectors:

      $$\vec{a_1} = \overrightarrow{A_1A_2} = \begin{pmatrix} 5-3 \\ 8-5 \\ 3-4 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}, \ \vec{a_2} = \overrightarrow{A_1A_3} = \begin{pmatrix} 1-3 \\ 2-5 \\ -2-4 \end{pmatrix} = \begin{pmatrix} -2 \\ -3 \\ -6 \end{pmatrix}.$$

      Thus, $$\vec{n} = \vec{a_1} \times \vec{a_2} = \begin{pmatrix} 18 \\ 8 \\ -6 \end{pmatrix}.$$

      The plane equation can be written as:

      $$18x + 8y - 6z + d = 0.$$

      In order to find $d$, we substitute the coordinates of point $A_1$:

      $$18 \cdot 3 + 8 \cdot 5 - 6 \cdot 4 + d = 0 \Rightarrow d = -6.$$

      Thus, the equation of the plane A1A2A3 has the form:

      $$18x + 8y - 6z - 6 = 0.$$

      In order to find the equation of straight line A1A2, you need to find its direction vector:

      $$\vec{b} = \overrightarrow{A_1A_2} = \begin{pmatrix} 5-3 \\ 8-5 \\ 3-4 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}.$$

      Thus, the equation of straight line A1A2 has the form:

      $$\begin{cases} x = 3 + 2t \\ y = 5 + 3t \\ z = 4 - t \end{p>

      In order to find the equation of the straight line A4M perpendicular to the plane A1A2A3, it is necessary to find the direction vector of this straight line. The direction vector will be directed along the vector perpendicular to the plane A1A2A3, and we have already found this vector when solving problem a). Thus, the direction vector of straight line A4M is equal to:

      $$\vec{c} = \vec{n} = \begin{pmatrix} 18 \\ 8 \\ -6 \end{pmatrix}.$$

      Considering that the point $M$ has coordinates $(1;-3;3)$, the equation of straight line A4M has the form:

      $$\begin{cases} x = 1 + 18t \\ y = -3 + 8t \\ z = 3 - 6t \end{cases}.$$

      In order to find the equation of straight line A3N parallel to straight line A1A2, it is necessary to find its direction vector. The direction vector of this line will be collinear to the vector directed along straight line A1A2, that is:

      $$\vec{d} = \overrightarrow{A_1A_2} = \begin{pmatrix} 5-3 \\ 8-5 \\ 3-4 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \\ -1 \end{pmatrix}.$$

      Thus, the equation of straight line A3N has the form:

      $$\begin{cases} x = 1 - 2t \\ y = 2 - 3t \\ z = -2 - t \end{cases}.$$

      In order to find the equation of a plane passing through point A4 and perpendicular to line A1A2, it is necessary to find its normal vector. The normal vector of the plane will be collinear to the vector product of the vector directed along straight line A1A2 and the vector directed from point A4 to the point of intersection of straight line A1A2 with the plane A1A2A3. Thus, the normal vector of the plane is:

      $$\vec{m} = \vec{b} \times (\vec{a_1} \times \vec{b}) = \begin{pmatrix} -10 \\ 14 \\ -6 \end{pmatrix}.$$

      Considering that point A4 has coordinates $(-1;0;2)$, the plane equation has the form:

      $$-10x + 14y - 6z + d = 0.$$

      In order to find $d$, we substitute the coordinates of point A4:

      $$-10 \cdot (-1) + 14 \cdot 0 - 6 \cdot 2 + d = 0 \Rightarrow d = -2.$$

      Thus, the equation of the plane passing through point A4 and perpendicular to line A1A2 has the form:

      $$-10x + 14y - 6z - 2 = 0.$$

      To calculate the sine of the angle between straight line A1A4 and plane A1A2A3

      IDZ Ryabushko 3.1 Option 3 is an educational and methodological complex intended for students and schoolchildren studying mathematics at the highest level. The complex contains detailed theoretical materials, examples of problem solving and practical tasks that will help deepen knowledge and skills in the field of mathematics.

      The complex includes sections on algebra, geometry and mathematical analysis, which allow you to cover a wide range of problems and topics. The complex also contains tasks of varying complexity, which allows students and schoolchildren to choose tasks depending on their level of preparation.

      Complex IDZ Ryabushko 3.1 Option 3 is presented in a beautiful html design, which makes it easier to work with materials and allows you to quickly find the information you need. The complex is easily downloaded and installed on a personal computer, tablet or smartphone, which makes it available for use at any convenient time.

      Ryabushko IDZ 3.1 Option 3 is an indispensable assistant for students and schoolchildren studying mathematics at the highest level, and allows you to effectively cope with difficult problems and deepen your knowledge in the field of mathematics.

IDZ Ryabushko 3.1 Option 3 is a math task that includes solving several problems on finding equations of planes and lines in three-dimensional space, as well as finding angles between lines and planes. The task is given four points in three-dimensional space and requires you to find various geometric objects associated with these points.

To solve problems in the task, it is necessary to use knowledge of vector and scalar algebra, as well as geometry in three-dimensional space. In solving each problem, it is necessary to consistently apply the studied formulas and solution methods in order to obtain answers to the questions posed.

Ryabushko IDZ complex 3.1 Option 3 can be useful for students and schoolchildren studying mathematics at the highest level to consolidate and deepen their knowledge and skills in three-dimensional geometry and vector algebra.

***

Assassin's Creed II STEAM•RU is an action-adventure computer game that is available for download on the Steam platform in Russia. Players take on the role of ?zio Auditore da Firenze and travel to 15th-century Italy to avenge the murder of their family and become part of a secret order of assassins.

The product also offers auto-shipping, which means the buyer will receive their product instantly after payment. In addition, there is a 10% discount on the product if you use your card when purchasing.

***

1. IDZ Ryabushko 3.1 Option 3 is an excellent digital product for preparing for the exam.
2. The convenient format and structure of Ryabushko IDZ 3.1 Option 3 allows you to quickly and efficiently repeat the material.
3. Solving tasks of IDZ Ryabushko 3.1 Option 3 helps to improve problem-solving skills and increase academic performance.
4. IDZ Ryabushko 3.1 Option 3 contains useful recommendations and tips for preparing for the exam.
5. A large number of tasks in Ryabushko IDZ 3.1 Option 3 allows you to cover a wide range of topics included in the exam paper.
6. IDZ Ryabushko 3.1 Option 3 is a great way to test your knowledge and prepare for the exam at a high level.
7. Ryabushko IDZ 3.1 Option 3 helps build confidence before the exam and reduces stress levels.
8. IDZ Ryabushko 3.1 Option 3 is a convenient and affordable digital product that can be used anywhere and anytime.
9. Ryabushko IDZ 3.1 Option 3 is an effective tool for improving academic performance and achieving better results in the exam.
10. IDZ Ryabushko 3.1 Option 3 is an excellent choice for those who want to prepare for the exam as effectively as possible and get a high grade.

Peculiarities:

A very convenient and practical digital product that helps you quickly and easily prepare for the exam.

IDZ Ryabushko 3.1 Option 3 contains useful materials and assignments for self-study.

This digital product is an excellent tool for improving knowledge and skills in mathematics.

In Ryabushko's IDZ 3.1 Option 3, mathematical concepts and principles are clearly and understandably explained.

The tasks in this digital product are well structured and cover a variety of topics.

Ryabushko IDZ 3.1 Option 3 contains many examples and tasks, which helps to better understand the material.

This digital product is ideal for those who want to prepare quickly and efficiently for their math exam.

IDZ Ryabushko 3.1 Option 3 is an excellent choice for those who want to improve their knowledge and skills in mathematics.

With this digital product, you can easily prepare for a math test or exam.

IDZ Ryabushko 3.1 Option 3 is an indispensable tool for everyone who studies mathematics and wants to improve their knowledge.