Consider in space some plane π and an arbitrary point M 0. Let us choose for the plane unit normal vector n with the beginning at some point М 1 ∈ π, and let p (М 0, π) be the distance from point М 0 to the plane π. Then (fig.5.5)
p (М 0, π) = | pr n M 1 M 0 | = | nM 1 M 0 |, (5.8)
since | n | = 1.
If the plane π is given in a rectangular coordinate system by its general equation Ax + By + Cz + D = 0, then its normal vector is a vector with coordinates (A; B; C) and as a unit normal vector one can choose
Let (x 0; y 0; z 0) and (x 1; y 1; z 1) coordinates of points M 0 and M 1. Then the equality Ax 1 + By 1 + Cz 1 + D = 0 holds, since the point M 1 belongs to the plane, and we can find the coordinates of the vector M 1 M 0: M 1 M 0 = (x 0 -x 1; y 0 -y 1; z 0 -z 1). Writing down scalar product nM 1 M 0 in coordinate form and transforming (5.8), we obtain
since Ax 1 + By 1 + Cz 1 = - D. So, to calculate the distance from a point to a plane, you need to substitute the coordinates of the point into the general equation of the plane, and then divide the absolute value of the result by a normalizing factor equal to the length of the corresponding normal vector.
PROBLEMS C2 OF THE UNIFIED STATE EXAM IN MATH FOR FINDING THE DISTANCE FROM A POINT TO A PLANE
Kulikova Anastasia Yurievna
5th year student, department of mat. analysis, algebra and geometry EI KFU, RF, Republic of Tatarstan, Elabuga
Ganeeva Aigul Rifovna
scientific adviser, Ph.D. ped. Sci., Associate Professor, EI KFU, RF, Republic of Tatarstan, Elabuga
V tasks of the exam in mathematics in last years there are tasks for calculating the distance from a point to a plane. In this article, using one task as an example, different methods finding the distance from a point to a plane. The most suitable method can be used to solve various problems. Having solved the problem with one method, another method can check the correctness of the result obtained.
Definition. The distance from a point to a plane that does not contain this point is the length of the perpendicular segment dropped from this point onto the given plane.
Task. Given a rectangular parallelepiped ABWITHDA 1 B 1 C 1 D 1 with sides AB=2, BC=4, AA 1 = 6. Find the distance from the point D to plane ASD 1 .
1 way. Using definition... Find the distance r ( D, ASD 1) from point D to plane ASD 1 (fig. 1).
Figure 1. First method
We will carry out DH⊥AS, therefore, by the theorem about three perpendiculars D 1 H⊥AS and (DD 1 H)⊥AS... We will carry out straight DT perpendicular D 1 H... Straight DT lies in the plane DD 1 H, hence DT⊥AC... Hence, DT⊥ASD 1.
ADC find the hypotenuse AS and height DH
From a right triangle D 1 DH find the hypotenuse D 1 H and height DT
Answer: .
Method 2.Volume method (using an auxiliary pyramid). The problem of this type can be reduced to the problem of calculating the height of a pyramid, where the height of the pyramid is the desired distance from a point to a plane. Prove that this height is the desired distance; find the volume of this pyramid in two ways and express this height.
Note that with this method there is no need to construct a perpendicular from a given point to a given plane.
Rectangular parallelepiped - a parallelepiped, all of whose faces are rectangles.
AB=CD=2, BC=AD=4, AA 1 =6.
The desired distance is the height h pyramids ACD 1 D dropped from the top D on the basis ACD 1 (fig. 2).
Let's calculate the volume of the pyramid ACD 1 D two ways.
Calculating, in the first way we take as the base ∆ ACD 1, then
Calculating, in the second way we take as the base ∆ ACD, then
Equating the right-hand sides of the last two equalities, we obtain
Figure 2. Second method
Of right-angled triangles ASD, ADD 1 , CDD 1 find the hypotenuses using the Pythagorean theorem
ACD
We calculate the area of the triangle ASD 1 using Heron's formula
Answer: .
Method 3. Coordinate method.
Let a point be given M(x 0 ,y 0 ,z 0) and the plane α given by the equation ax+by+cz+d= 0 in a rectangular Cartesian coordinate system. Distance from point M to the plane α can be calculated by the formula:
Let's introduce a coordinate system (Fig. 3). Origin of coordinates at a point V;
Straight AB- axis X, straight Sun- axis y, straight BB 1 - axis z.
Figure 3. The third method
B(0,0,0), A(2,0,0), WITH(0,4,0), D(2,4,0), D 1 (2,4,6).
Let ax +by+ cz+ d= 0 - plane equation ACD one . Substituting the coordinates of the points into it A, C, D 1 we get:
Plane equation ACD 1 will take the form
Answer: .
Method 4. Vector method.
Let us introduce a basis (Fig. 4),.
Figure 4. Fourth method
, Competition "Presentation for the lesson"
Class: 11
Lesson presentation
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.
Goals:
- generalization and systematization of knowledge and skills of students;
- development of skills to analyze, compare, draw conclusions.
Equipment:
- multimedia projector;
- computer;
- worksheets with texts of tasks
PROCESS OF THE LESSON
I. Organizational moment
II. Knowledge update stage(slide 2)
We repeat how the distance from a point to a plane is determined
III. Lecture(slides 3-15)
In this lesson, we will look at various ways to find the distance from a point to a plane.
First method: step-by-step computational
Distance from point M to plane α:
- equal to the distance to the plane α from an arbitrary point P lying on the straight line a, which passes through the point M and is parallel to the plane α;
- is equal to the distance to the plane α from an arbitrary point P lying on the plane β, which passes through the point M and is parallel to the plane α.
Let's solve the following tasks:
№1. In the cube A ... D 1 find the distance from point C 1 to plane AB 1 C.
It remains to calculate the value of the length of the segment O 1 N.
№2. In a regular hexagonal prism A ... F 1, all edges of which are equal to 1, find the distance from point A to the plane DEA 1.
The next method: volume method.
If the volume of the pyramid ABCM is equal to V, then the distance from point M to the plane α containing ∆ABS is calculated by the formula ρ (M; α) = ρ (M; ABC) =
When solving problems, we use the equality of the volumes of one figure, expressed in two different ways.
Let's solve the following problem:
№3. The edge AD of the pyramid DABC is perpendicular to the plane of the base ABC. Find the distance from A to the plane passing through the midpoints of the ribs AB, AC and AD, if.
When solving problems coordinate method the distance from point M to plane α can be calculated by the formula ρ (M; α) = 
, where M (x 0; y 0; z 0), and the plane is given by the equation ax + by + cz + d = 0
Let's solve the following problem:
№4. In the unit cube A ... D 1 find the distance from point A1 to plane BDC 1.
We introduce a coordinate system with the origin at point A, the y-axis will run along the AB edge, the x-axis along the AD edge, and the z-axis along the AA 1 edge. Then the coordinates of points B (0; 1; 0) D (1; 0; 0;) C 1 (1; 1; 1)
Let's compose the equation of the plane passing through the points B, D, C 1.
Then - dx - dy + dz + d = 0 x + y - z - 1 = 0. Therefore, ρ = 
The next method that can be used when solving problems of this type is - method of support tasks.
The application of this method consists in the application of known support problems, which are formulated as theorems.
Let's solve the following problem:
№5. In the unit cube A ... D 1 find the distance from point D 1 to plane AB 1 C.
Consider the application vector method.
№6. In the unit cube A ... D 1 find the distance from point A 1 to the plane BDC 1.
So, we looked at various methods that can be used to solve this type of problem. The choice of this or that method depends on the specific task and your preferences.
IV. Working in groups
Try to solve the problem in different ways.
№1. The edge of the cube A ... D 1 is equal. Find the distance from vertex C to plane BDC 1.
№2. V regular tetrahedron ABCD with edge find the distance from point A to plane BDC
№3. In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance from A to the plane BCA 1.
№4. In a regular rectangular pyramid SABCD with all edges equal to 1, find the distance from A to the SCD plane.
V. Lesson summary, homework, reflection
Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify a specific person or contact him.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information we collect:
- When you leave a request on the site, we may collect various information, including your name, phone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you and report unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notifications and messages.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, competition, or similar promotional event, we may use the information you provide to administer those programs.
Disclosure of information to third parties
We do not disclose information received from you to third parties.
Exceptions:
- If it is necessary - in accordance with the law, court order, in court proceedings, and / or on the basis of public requests or requests from government authorities on the territory of the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other socially important reasons.
- In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the appropriate third party - the legal successor.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and abuse, as well as from unauthorized access, disclosure, alteration and destruction.
Respect for your privacy at the company level
In order to make sure that your personal information is safe, we bring the rules of confidentiality and security to our employees, and strictly monitor the implementation of confidentiality measures.