The distance between skew lines is the length of their common perpendicular (the segment with ends on these lines and perpendicular to each of them). Stepwise computational method (construction of a common perpendicular). b ρ Example a

Construct a plane containing one of the lines and parallel to the other. Then the desired distance will be equal to the distance from some point of the second straight line to the constructed plane (at this stage, you can use coordinate method) Method of parallel lines and planes. Example b ρ a α A B shah.ucoz.ru/load/egeh/egeh_s2/k oordinatnyj_metod_kljuchevye_za dachi/

Construct a plane perpendicular to one of the given lines, and construct an orthogonal projection of the other line on this plane. Orthogonal design method. Example b ρ a α A B L N S NE - projection b

If AB and CD are crossing edges of the triangular pyramid ABCD, d is the distance between them, α is the angle between AB and CD, V is the volume of the pyramid ABCD, then the Support Problem. Example B C A D For methods of finding the angle between lines, see:

From the system, determine the coordinates, then find Let, then the condition is satisfied: Determine the coordinates of the direction vectors and. Vector - coordinate method. Example B C A D Note: to record the coordinates of points M and K, use the formula: M K

In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the distance between lines BD and SA. Solution: D. p .: OH can be found from the triangle AOS by the area method. O A B C D S H OH - common perpendicular to lines BD and AS Back

In a regular quadrangular pyramid SABCD, all edges of which are equal to 1, find the distance between lines BD and SA. Solution: (half the diagonal of the unit square) O A B C D S H Back

In the right triangular prism ABCA 1 C 1 B 1, all edges of which are equal to 1, find the distance between the lines AA 1 and B 1 C. Solution: B C C1C1 B1B1 H A A1A1 D. p .: (perpendicular drawn to the intersection of perpendicular planes) From triangle ASN Back

In a regular truncated quadrangular pyramid ABCDA 1 B 1 C 1 D 1 with base sides equal to 4 and 8 and height equal to 6, find the distance between the diagonal and BD 1 of the diagonal of the larger base AC. Solution: B A C D A1A1 B1B1 C1C1 D1D1 O O1O1 D.p.: H (is its projection onto (BB 1 D 1)) Consider isosceles trapezoid BB 1 D 1 D Back

In a regular truncated quadrangular pyramid ABCDA 1 B 1 C 1 D 1 with base sides equal to 4 and 8 and height equal to 6, find the distance between the diagonal and BD 1 of the diagonal of the larger base AC. Solution: BD B1B1 D1D1 O Back K H In triangle BD 1 K Triangles BD 1 K and BON are similar in two angles In triangle BHO

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube BD 1 and the diagonal of the face AB 1. Solution: Consider the pyramid D 1 AB 1 B. Take AB 1 B as the base, then the height is BC. (diagonal of a unit square) A C D D1D1 B1B1 C A1A1 B (diagonal of a unit cube) Find the angle between the lines AB 1 and B 1 D 1. You can use the vector-coordinate method. Back

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube BD 1 and the diagonal of the face AB 1. Solution: Let's introduce a rectangular coordinate system A C D D1D1 B1B1 C A1A1 B X Z Y Then: Back

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube BD 1 and the diagonal of the face AB 1. Solution: A C D D1D1 B1B1 C A1A1 B Back

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube AB 1 and the diagonal of the face A 1 C 1. Solution: A C D D1D1 B1B1 C A1A1 B We introduce a rectangular coordinate system Then: Let M K Then: X Z Y Back and

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube AB 1 and the diagonal of the face A 1 C 1. Solution: A C D D1D1 B1B1 C A1A1 B X Z Y M K Back

In the unit cube ABCDA 1 B 1 C 1 D 1 find the distance between the diagonal of the cube AB 1 and the diagonal of the face A 1 C 1. Solution: Back

2) In a regular quadrangular pyramid MABCD, all edges of which are equal to 1, find the distance between lines MA and BC Training exercises Solution 3) The side of the base ABC of a regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between lines BA 1 and AC 1. Solution 1) Find the distance between non-intersecting diagonals of two adjacent faces of a cube whose edge length is 1.

Solution: Back Problems 1) Find the distance between non-intersecting diagonals of two adjacent faces of a cube whose edge length is 1. A C D D1D1 B1B1 C A1A1 B O O O1O1 H We construct an orthogonal projection of the line AB 1 onto the plane (BB 1 D 1) D. p .: Find O 1 H find from the triangle B 1 OO 1

Solution: A D B C M O N 2) In a regular quadrangular pyramid MABCD, all edges of which are equal to 1, find the distance between lines MA and BC. (triangle AMD is equilateral) Find the angle between lines AD and BC. Tasks of the Armed Forces || AD => "> "> " title="(!LANG:Solution: A D B C M O N 2) In a regular quadrangular pyramid MABCD with all edges equal to 1, find the distance between lines MA and BC. (triangle AMD is equilateral) Find the angle between lines AD and BC. Tasks of the Armed Forces || AD =>"> title="Solution: A D B C M O N 2) In a regular quadrangular pyramid MABCD, all edges of which are equal to 1, find the distance between lines MA and BC. (triangle AMD is equilateral) Find the angle between lines AD and BC. Tasks of the Armed Forces || AD =>"> !}

A B C D Solution: A1A1 C1C1 3) The side of the base ABC of the regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between the lines BA 1 and AC 1. The segments AC 1 and BA 1 are the edges of the triangular pyramid C 1 ABA 1 (support problem). 5) The volume of the pyramid with the base VA 1 A? 4) Distance from point C 1 to the plane (BDA) (height of the pyramid)? 6) ρ(VA 1 ;AC 1)? 1) The lengths of the ribs BA 1 and AC 1? 2) The sine of the angle between the straight lines BA 1 and AC 1? 3) The area of ​​the base of the pyramid - VA 1 A? O Tasks

A 3) The side of the base ABC of the regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between the lines BA 1 and AC 1. Solution: O A D A1A1 X Z Y x CxC 1) Introduce a rectangular coordinate system Then: xDxD Find the coordinates of points C and D B X Y O C H (triangle median property) xDxD x CxC C B C1C1 Tasks

The side of the base ABC of the regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between lines BA 1 and AC 1. Solution: A B C D A1A1 C1C1 X Z Y (midpoints of CD and AD) Determine the coordinates of the direction vectors Problems

The side of the base ABC of the regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between lines BA 1 and AC 1. Solution: 4) Find the distance from point C 1 to the plane (BDA) (height of the pyramid). Let's derive the equation of the plane (EFP) Problems

A B C D Solution: A1A1 C1C1 3) The side of the base ABC of the regular triangular pyramid ABCD is equal, the height of the pyramid is DO=6. Points A 1, C 1 are the midpoints of the edges AD and CD, respectively. Find the distance between lines BA 1 and AC 1. 5) Find the volume of the pyramid with base BA 1 A? O Tasks

When creating the presentation, the manual was used:

"Distance between skew lines" - Theorem. Preparatory oral tasks. Find the distance between the line MN and the plane AA1D1D. Find the distance between line B1K and plane DD1C1C. OK=OO1?OM/O1M =a/3 (according to the Pythagorean theorem O1M=3/2?2, OM=1/2?2). Diagonal plane AA1C1C is perpendicular to line BD. The new positions of points B and N will be the points of lines AD and BM closest to each other.

"Lesson Speed ​​time distance" - Mathematical warm-up. The purpose of the lesson: to teach students to solve problems on the movement. Distance. How long does it take to walk 30 km at a constant speed of 5 km/h? Relationship between speed, time and distance. How many people went to the city? An airplane flies the distance from city A to city B in 1 hour and 20 minutes.

“Speed ​​time distance mathematics” - Reduce the sum of the numbers 5 and 65 by 2 times. Dunno went to the moon. Journey through the pages of a fairy tale book. Fizkultminutka. One left at 8 o'clock and the other at 10 o'clock. Summarizing. Is Laura right? -Laura solved the following problem: “500 km. A car will pass in 10 hours. Time. The key with the answer "38" opens the book:

"Dialogue direct speech" - What is the difference between direct speech and dialogue? For example: L. N. Tolstoy said: “We all need each other in the world.” Graphics of direct speech. A: "p." Task 3. Replace direct speech with dialogue. For example: "P?" - but. "P!" - but. Point out the correct diagrams for the following sentences. Dialog graphics. How to write direct speech and dialogue in writing?

"Sentences with direct speech" - Petronius, ancient Roman writer. Game "Find the mistake" (check). Author's words introducing direct speech: I reappeared and went to the house of Father Gerasim. A friend from the village came to visit me. Proposals with direct speech. Creative task. In writing, direct speech is enclosed in quotation marks. Read!" exclaimed Konstantin Georgievich Paustovsky.

"Distance and Scale" - Model of the atom in high magnification scale. On a map with a scale, the distance is 5 cm. If the scale is given by a fraction with a numerator of 1, then. Scale model of a fire engine. Algorithm for finding the distance on the ground: On the highway, the length of the route is 700 km. Finish the sentence: The distance between two cities is 400 km.

DISTANCE BETWEEN RIGHTS IN SPACE The distance between two intersecting lines in space is the length of the common perpendicular drawn to these lines. If one of the two intersecting lines lies in a plane, and the other is parallel to this plane, then the distance between the given lines is equal to the distance between the line and the plane. If two intersecting lines lie in parallel planes, then the distance between these lines is equal to the distance between parallel planes.

Cube 1 In the unit cube A…D 1 find the distance between lines AA 1 and BC. Answer: 1.

Cube 2 In the unit cube A…D 1 find the distance between lines AA 1 and CD. Answer: 1.

Cube 3 In the unit cube A…D 1 find the distance between lines AA 1 and B 1 C 1. Answer: 1.

Cube 4 In the unit cube A…D 1 find the distance between lines AA 1 and C 1 D 1. Answer: 1.

Cube 5 In the unit cube A…D 1 find the distance between lines AA 1 and BC 1. Answer: 1.

Cube 6 In the unit cube A…D 1 find the distance between lines AA 1 and B 1 C. Answer: 1.

Cube 7 In the unit cube A…D 1 find the distance between lines AA 1 and CD 1. Answer: 1.

Cube 8 In the unit cube A…D 1 find the distance between lines AA 1 and DC 1. Answer: 1.

Cube 9 In the unit cube A…D 1 find the distance between lines AA 1 and CC 1. Answer:

Cube 10 In the unit cube A…D 1 find the distance between lines AA 1 and BD. Solution. Let O be the midpoint of BD. The desired distance is the length of the segment AO. It is equal to Answer:

Cube 11 In the unit cube A…D 1 find the distance between lines AA 1 and B 1 D 1. Answer:

Cube 12 In the unit cube A…D 1 find the distance between lines AA 1 and BD 1. Solution. Let P, Q be the midpoints of AA 1, BD 1. The desired distance is the length of the segment PQ. It is equal to Answer:

Cube 13 In the unit cube A…D 1 find the distance between lines AA 1 and BD 1. Answer:

Cube 14 In the unit cube A…D 1 find the distance by lines AB 1 and CD 1. Answer: 1.

Cube 15 In the unit cube A…D 1 find the distance between lines AB 1 and BC 1. Solution. The desired distance is equal to the distance between the parallel planes AB 1 D 1 and BDC 1. The diagonal A 1 C is perpendicular to these planes and is divided into three equal parts at the intersection points. Therefore, the desired distance is equal to the length of the segment EF and is equal to Answer:

Cube 16 In the unit cube A…D 1 find the distance between the lines AB 1 and A 1 C 1. The solution is similar to the previous one. Answer:

Cube 17 In the unit cube A…D 1 find the distance between lines AB 1 and BD. The solution is similar to the previous one. Answer:

Cube 18 In the unit cube A…D 1 find the distance by lines AB 1 and BD 1. Solution. Diagonal BD 1 is perpendicular to the plane equilateral triangle ACB 1 and intersects it at the center P of the circle inscribed in it. The desired distance is equal to the radius OP of this circle. OP = Answer:

Pyramid 1 In unit tetrahedron ABCD find the distance between lines AD and BC. Solution. The desired distance is equal to the length of the segment EF, where E, F are the midpoints of the edges AD, GF. In triangle DAG DA = 1, AG = DG = Answer: Therefore, EF =

Pyramid 2 In a regular pyramid SABCD, all edges of which are equal to 1, find the distance between lines AB and CD. Answer: 1.

Pyramid 3 In a regular pyramid SABCD, all edges of which are equal to 1, find the distance between lines SA and BD. Solution. The desired distance is equal to the height OH of the triangle SAO, where O is the midpoint of BD. IN right triangle SAO we have: SA = 1, AO = SO = Answer: Therefore, OH =

Pyramid 4 In a regular pyramid SABCD, all edges of which are equal to 1, find the distance between lines SA and BC. Solution. Plane SAD is parallel to line BC. Therefore, the desired distance is equal to the distance between the line BC and the plane SAD. It is equal to the height EH of the triangle SEF, where E, F are the midpoints of the edges BC, AD. In triangle SEF we have: EF = 1, SE = SF = Height SO is Therefore, EH = Answer:

Pyramid 5 In a regular 6th pyramid SABCDEF with base edges equal to 1, find the distance between lines AB and DE. Answer:

Pyramid 6 In the regular 6th pyramid SABCDEF, whose side edges are 2 and the base edges are 1, find the distance between lines SA and BC. Solution: Extend edges BC and AF until they intersect at point G. The common perpendicular to SA and BC is the altitude AH of triangle ABG. It is equal to Answer:

Pyramid 7 In the regular 6th pyramid SABCDEF, whose side edges are 2 and the base edges are 1, find the distance between lines SA and BF. Solution: The desired distance is the height GH of the triangle SAG, where G is the intersection point of BF and AD. In the triangle SAG we have: SA = 2, AG = 0.5, height SO is equal to From here we find GH = Answer:

Pyramid 8 In the regular 6th pyramid SABCDEF, whose side edges are 2 and the base edges are 1, find the distance between lines SA and CE. Solution: The desired distance is the height GH of the triangle SAG, where G is the intersection point of CE and AD. In the triangle SAG we have: SA = 2, AG = , the height SO is equal to From here we find GH = Answer:

Pyramid 9 In the regular 6th pyramid SABCDEF, whose side edges are 2 and the base edges are 1, find the distance between lines SA and BD. Solution: Line BD is parallel to plane SAE. The desired distance is equal to the distance between the line BD and this plane and is equal to the height PH of the triangle SPQ. In this triangle, the height SO is, PQ = 1, SP = SQ = From here we find PH = Answer:

Pyramid 10 In the regular 6th pyramid SABCDEF, whose lateral edges are 2 and the base edges are 1, find the distance between lines SA and BG, where G is the midpoint of edge SC. Solution: Draw a line through point G parallel to SA. Let Q denote the point of its intersection with the line AC. The desired distance is equal to the height QH of the right triangle ASQ, in which AS = 2, AQ = , SQ = From here we find QH = Answer: .

Prism 1 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: BC and B 1 C 1. Answer: 1.

Prism 2 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: AA 1 and BC. Answer:

Prism 3 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: AA 1 and BC 1. Answer:

Prism 4 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: AB and A 1 C 1. Answer: 1.

Prism 5 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: AB and A 1 C. Solution: The desired distance is equal to the distance between the line AB and the plane A 1 B 1 C. Let us denote D and D 1 the midpoints of the edges AB and A 1 B 1. In a right triangle CDD 1, draw a height DE from the vertex D. It will be the desired distance. We have, DD 1 = 1, CD = Answer: Therefore, DE = , CD 1 = .

Prism 6 In a regular triangular prism ABCA 1 B 1 C 1, all edges of which are equal to 1, find the distance between the lines: AB 1 and BC 1. Solution: Let's build the prism to a 4-angled prism. The desired distance will be equal to the distance between the parallel planes AB 1 D 1 and BDC 1. It is equal to the height OH of the right triangle AOO 1, in which the Answer. This height is

Prism 7 In the correct 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB and A 1 B 1. Answer: 1.

Prism 8 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB and B 1 C 1. Answer: 1.

Prism 9 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB and C 1 D 1. Answer: 1.

Prism 10 In the correct 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB and DE. Answer: .

Prism 11 In the correct 6th prism A ... F 1, whose edges are equal to 1, find the distance between the lines: AB and D 1 E 1. Answer: 2.

Prism 12 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and CC 1. Answer: .

Prism 13 In the correct 6th prism A ... F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and DD 1. Answer: 2.

Prism 14 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and B 1 C 1. Solution: Let's continue the sides B 1 C 1 and A 1 F 1 until they intersect at point G. Triangle A 1 B 1 G is equilateral. Its height A 1 H is the desired common perpendicular. Its length is equal. Answer: .

Prism 15 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and C 1 D 1. Solution: The desired common perpendicular is the segment A 1 C 1. Its length is equal. Answer: .

Prism 16 In the correct 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and BC 1. Solution: The desired distance is the distance between the parallel planes ADD 1 and BCC 1. It is equal. Answer: .

Prism 17 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and CD 1. Solution: The desired common perpendicular is the segment AC. Its length is equal. Answer: .

Prism 18 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and DE 1. Solution: The desired common perpendicular is the segment A 1 E 1. Its length is equal. Answer: .

Prism 19 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and BD 1. Solution: The desired common perpendicular is the segment AB. Its length is 1. Answer: 1.

Prism 20 In a regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and CE 1. Solution: The desired distance is the distance between the line AA 1 and the plane CEE 1. It is equal. Answer: .

Prism 21 In the correct 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and BE 1. Solution: The desired distance is the distance between the line AA 1 and the plane BEE 1. It is equal. Answer: .

Prism 22 In a regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AA 1 and CF 1. Solution: The desired distance is the distance between the line AA 1 and the plane CFF 1. It is equal. Answer: .

Prism 23 In the regular 6th prism A…F 1, whose edges are equal to 1, find the angle between the lines: AB 1 and DE 1. Solution: The desired distance is the distance between the parallel planes ABB 1 and DEE 1. The distance between them is equal. Answer: .

Prism 24 In a regular 6th prism A…F 1, whose edges are equal to 1, find the angle between the lines: AB 1 and CF 1. Solution: The desired distance is the distance between the line AB 1 and the plane CFF 1. It is equal. Answer:

Prism 25 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB 1 and BC 1. Solution: Let O, O 1 be the centers of the faces of the prism. Planes AB 1 O 1 and BC 1 O are parallel. The plane ACC 1 A 1 is perpendicular to these planes. The desired distance d is equal to the distance between the lines AG 1 and GC 1. In the parallelogram AGC 1 G 1 we have AG = Answer: ; AG 1 = The height drawn to the side AA 1 is equal to 1. Therefore, d= . .

Prism 26 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB 1 and BD 1. Solution: Consider the plane A 1 B 1 HG perpendicular to BD 1. The orthogonal projection onto this plane translates the line BD 1 to point H, and line AB 1 to line GB 1. Therefore, the desired distance d is equal to the distance from point H to line GB 1. In a right triangle GHB 1 we have GH = 1; Answer: B 1 H = . Therefore, d = .

Prism 27 In the regular 6th prism A…F 1, whose edges are equal to 1, find the distance between the lines: AB 1 and BE 1. Solution: Consider the plane A 1 BDE 1, perpendicular to AB 1. The orthogonal projection onto this plane translates the line AB 1 to point G, and the line BE 1 leaves in place. Therefore, the desired distance d is equal to the distance GH from the point G to the line BE 1. In a right triangle A 1 BE 1 we have A 1 B = ; A 1 E 1 =. Answer: Therefore, d = .

In this article, using the example of solving problem C2 from the Unified State Examination, the method of finding coordinates using the method is analyzed. Recall that lines are skew if they do not lie in the same plane. In particular, if one line lies in a plane, and the second line intersects this plane at a point that does not lie on the first line, then such lines are skew (see figure).

For finding distances between intersecting lines necessary:

1. Draw a plane through one of the skew lines that is parallel to the other skew line.
2. Drop a perpendicular from any point of the second straight line to the resulting plane. The length of this perpendicular will be the desired distance between the lines.

Let us analyze this algorithm in more detail using the example of solving problem C2 from the Unified State Examination in mathematics.

Distance between lines in space

A task. in a single cube ABCDA 1 B 1 C 1 D 1 find the distance between the lines BA 1 and D.B. 1 .

Rice. 1. Drawing for the task

Solution. Through the midpoint of the diagonal of the cube D.B. 1 (dot O) draw a line parallel to the line A 1 B. Points of intersection of a given line with edges BC And A 1 D 1 denote respectively N And M. Straight MN lies in the plane MNB 1 and parallel to the line A 1 B, which does not lie in this plane. This means that the direct A 1 B parallel to the plane MNB 1 on the basis of parallelism of a straight line and a plane (Fig. 2).

Rice. 2. The desired distance between the crossing lines is equal to the distance from any point of the selected line to the depicted plane

We are now looking for the distance from some point on the straight line A 1 B up to the plane MNB one . This distance, by definition, will be the desired distance between the skew lines.

To find this distance, we use the coordinate method. We introduce a rectangular Cartesian coordinate system so that its origin coincides with point B, the axis X was directed along the edge BA, axis Y- along the rib BC, axis Z- along the rib BB 1 (Fig. 3).

Rice. 3. We choose a rectangular Cartesian coordinate system as shown in the figure

We find the equation of the plane MNB 1 in this coordinate system. To do this, we first determine the coordinates of the points M, N And B 1: The resulting coordinates are substituted into general equation straight line and we get the following system of equations:

From the second equation of the system, we obtain from the third one, and then from the first we obtain. We substitute the obtained values ​​into the general equation of the straight line:

Note that otherwise the plane MNB 1 would pass through the origin. We divide both sides of this equation by and we get:

The distance from a point to a plane is determined by the formula.