It will be useful for every student who is preparing for the exam in mathematics to repeat the topic “Finding the angle between lines”. As statistics show, when passing an attestation test, tasks in this section of stereometry cause difficulties for a large number of students. At the same time, tasks requiring finding the angle between straight lines are found in the USE at both the basic and profile levels. This means that everyone should be able to solve them.

Basic moments

There are 4 types of mutual arrangement of lines in space. They can coincide, intersect, be parallel or intersecting. The angle between them can be acute or straight.

To find the angle between the lines in the Unified State Examination or, for example, in the solution, schoolchildren in Moscow and other cities can use several methods for solving problems in this section of stereometry. You can complete the task by classical constructions. To do this, it is worth learning the basic axioms and theorems of stereometry. The student needs to be able to logically build reasoning and create drawings in order to bring the task to a planimetric problem.

You can also use the vector-coordinate method, using simple formulas, rules and algorithms. The main thing in this case is to correctly perform all the calculations. The Shkolkovo educational project will help you hone your skills in solving problems in stereometry and other sections of the school course.

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Oh-oh-oh-oh-oh ... well, it's tinny, as if you read the sentence to yourself =) However, then relaxation will help, especially since I bought suitable accessories today. Therefore, let's proceed to the first section, I hope, by the end of the article I will keep a cheerful mood.

Mutual arrangement of two straight lines

The case when the hall sings along in chorus. Two lines can:

1) match;

2) be parallel: ;

3) or intersect at a single point: .

Help for dummies : please remember the mathematical sign of the intersection , it will occur very often. The entry means that the line intersects with the line at the point.

How to determine the relative position of two lines?

Let's start with the first case:

Two lines coincide if and only if their respective coefficients are proportional, that is, there is such a number "lambda" that the equalities

Let's consider straight lines and compose three equations from the corresponding coefficients: . From each equation it follows that, therefore, these lines coincide.

Indeed, if all the coefficients of the equation multiply by -1 (change signs), and all the coefficients of the equation reduce by 2, you get the same equation: .

The second case when the lines are parallel:

Two lines are parallel if and only if their coefficients at the variables are proportional: , but.

As an example, consider two straight lines. We check the proportionality of the corresponding coefficients for the variables :

However, it is clear that .

And the third case, when the lines intersect:

Two lines intersect if and only if their coefficients of the variables are NOT proportional, that is, there is NOT such a value of "lambda" that the equalities are fulfilled

So, for straight lines we will compose a system:

From the first equation it follows that , and from the second equation: , hence, the system is inconsistent(no solutions). Thus, the coefficients at the variables are not proportional.

Conclusion: lines intersect

In practical problems, the solution scheme just considered can be used. By the way, it is very similar to the algorithm for checking vectors for collinearity, which we considered in the lesson. The concept of linear (non) dependence of vectors. Vector basis. But there is a more civilized package:

Example 1

Find out the relative position of the lines:

Decision based on the study of directing vectors of straight lines:

a) From the equations we find the direction vectors of the lines: .

, so the vectors are not collinear and the lines intersect.

Just in case, I will put a stone with pointers at the crossroads:

The rest jump over the stone and follow on, straight to Kashchei the Deathless =)

b) Find the direction vectors of the lines:

The lines have the same direction vector, which means they are either parallel or the same. Here the determinant is not necessary.

Obviously, the coefficients of the unknowns are proportional, while .

Let's find out if the equality is true:

Thus,

c) Find the direction vectors of the lines:

Let's calculate the determinant, composed of the coordinates of these vectors:
, therefore, the direction vectors are collinear. The lines are either parallel or coincide.

The proportionality factor "lambda" is easy to see directly from the ratio of collinear direction vectors. However, it can also be found through the coefficients of the equations themselves: .

Now let's find out if the equality is true. Both free terms are zero, so:

The resulting value satisfies this equation (any number generally satisfies it).

Thus, the lines coincide.

Answer:

Very soon you will learn (or even have already learned) to solve the considered problem verbally literally in a matter of seconds. In this regard, I see no reason to offer something for an independent solution, it is better to lay one more important brick in the geometric foundation:

How to draw a line parallel to a given one?

For ignorance of this simplest task, the Nightingale the Robber severely punishes.

Example 2

The straight line is given by the equation . Write an equation for a parallel line that passes through the point.

Decision: Denote the unknown line by the letter . What does the condition say about it? The line passes through the point. And if the lines are parallel, then it is obvious that the directing vector of the line "ce" is also suitable for constructing the line "te".

We take out the direction vector from the equation:

Answer:

The geometry of the example looks simple:

Analytical verification consists of the following steps:

1) We check that the lines have the same direction vector (if the equation of the line is not properly simplified, then the vectors will be collinear).

2) Check if the point satisfies the resulting equation.

Analytical verification in most cases is easy to perform orally. Look at the two equations and many of you will quickly figure out how the lines are parallel without any drawing.

Examples for self-solving today will be creative. Because you still have to compete with Baba Yaga, and she, you know, is a lover of all kinds of riddles.

Example 3

Write an equation for a line passing through a point parallel to the line if

There is a rational and not very rational way to solve. The shortest way is at the end of the lesson.

We did a little work with parallel lines and will return to them later. The case of coinciding lines is of little interest, so let's consider a problem that is well known to you from the school curriculum:

How to find the point of intersection of two lines?

If straight intersect at the point , then its coordinates are the solution systems of linear equations

How to find the point of intersection of lines? Solve the system.

Here's to you geometric meaning of a system of two linear equations with two unknowns are two intersecting (most often) straight lines on a plane.

Example 4

Find the point of intersection of lines

Decision: There are two ways to solve - graphical and analytical.

The graphical way is to simply draw the given lines and find out the point of intersection directly from the drawing:

Here is our point: . To check, you should substitute its coordinates into each equation of a straight line, they should fit both there and there. In other words, the coordinates of a point are the solution of the system . In fact, we considered a graphical way to solve systems of linear equations with two equations, two unknowns.

The graphical method, of course, is not bad, but there are noticeable disadvantages. No, the point is not that seventh graders decide this way, the point is that it will take time to make a correct and EXACT drawing. In addition, some lines are not so easy to construct, and the intersection point itself may be somewhere in the thirtieth kingdom outside the notebook sheet.

Therefore, it is more expedient to search for the intersection point by the analytical method. Let's solve the system:

To solve the system, the method of termwise addition of equations was used. To develop the relevant skills, visit the lesson How to solve a system of equations?

Answer:

The verification is trivial - the coordinates of the intersection point must satisfy each equation of the system.

Example 5

Find the point of intersection of the lines if they intersect.

This is a do-it-yourself example. It is convenient to divide the problem into several stages. Analysis of the condition suggests that it is necessary:
1) Write the equation of a straight line.
2) Write the equation of a straight line.
3) Find out the relative position of the lines.
4) If the lines intersect, then find the point of intersection.

The development of an action algorithm is typical for many geometric problems, and I will repeatedly focus on this.

Full solution and answer at the end of the tutorial:

A pair of shoes has not yet been worn out, as we got to the second section of the lesson:

Perpendicular lines. The distance from a point to a line.
Angle between lines

Let's start with a typical and very important task. In the first part, we learned how to build a straight line parallel to the given one, and now the hut on chicken legs will turn 90 degrees:

How to draw a line perpendicular to a given one?

Example 6

The straight line is given by the equation . Write an equation for a perpendicular line passing through a point.

Decision: It is known by assumption that . It would be nice to find the direction vector of the straight line. Since the lines are perpendicular, the trick is simple:

From the equation we “remove” the normal vector: , which will be the directing vector of the straight line.

We compose the equation of a straight line by a point and a directing vector:

Answer:

Let's unfold the geometric sketch:

Hmmm... Orange sky, orange sea, orange camel.

Analytical verification of the solution:

1) We extract the direction vectors from the equations and use dot product of vectors we conclude that the lines are indeed perpendicular: .

By the way, you can use normal vectors, it's even easier.

2) Check if the point satisfies the resulting equation .

Verification, again, is easy to perform verbally.

Example 7

Find the point of intersection of perpendicular lines, if the equation is known and dot.

This is a do-it-yourself example. There are several actions in the task, so it is convenient to arrange the solution point by point.

Our exciting journey continues:

Distance from point to line

Before us is a straight strip of the river and our task is to reach it in the shortest way. There are no obstacles, and the most optimal route will be movement along the perpendicular. That is, the distance from a point to a line is the length of the perpendicular segment.

The distance in geometry is traditionally denoted by the Greek letter "ro", for example: - the distance from the point "em" to the straight line "de".

Distance from point to line is expressed by the formula

Example 8

Find the distance from a point to a line

Decision: all you need is to carefully substitute the numbers into the formula and do the calculations:

Answer:

Let's execute the drawing:

The distance found from the point to the line is exactly the length of the red segment. If you make a drawing on checkered paper on a scale of 1 unit. \u003d 1 cm (2 cells), then the distance can be measured with an ordinary ruler.

Consider another task according to the same drawing:

The task is to find the coordinates of the point , which is symmetrical to the point with respect to the line . I propose to perform the actions on your own, however, I will outline the solution algorithm with intermediate results:

1) Find a line that is perpendicular to a line.

2) Find the point of intersection of the lines: .

Both actions are discussed in detail in this lesson.

3) The point is the midpoint of the segment. We know the coordinates of the middle and one of the ends. By formulas for the coordinates of the middle of the segment find .

It will not be superfluous to check that the distance is also equal to 2.2 units.

Difficulties here may arise in calculations, but in the tower a microcalculator helps out a lot, allowing you to count ordinary fractions. Have advised many times and will recommend again.

How to find the distance between two parallel lines?

Example 9

Find the distance between two parallel lines

This is another example for an independent solution. A little hint: there are infinitely many ways to solve. Debriefing at the end of the lesson, but better try to guess for yourself, I think you managed to disperse your ingenuity well.

Angle between two lines

Whatever the corner, then the jamb:


In geometry, the angle between two straight lines is taken as the SMALLER angle, from which it automatically follows that it cannot be obtuse. In the figure, the angle indicated by the red arc is not considered to be the angle between intersecting lines. And its “green” neighbor or oppositely oriented crimson corner.

If the lines are perpendicular, then any of the 4 angles can be taken as the angle between them.

How are the angles different? Orientation. First, the direction of "scrolling" the corner is fundamentally important. Secondly, a negatively oriented angle is written with a minus sign, for example, if .

Why did I say this? It seems that you can get by with the usual concept of an angle. The fact is that in the formulas by which we will find the angles, a negative result can easily be obtained, and this should not take you by surprise. An angle with a minus sign is no worse, and has a very specific geometric meaning. In the drawing for a negative angle, it is imperative to indicate its orientation (clockwise) with an arrow.

How to find the angle between two lines? There are two working formulas:

Example 10

Find the angle between lines

Decision and Method one

Consider two straight lines given by equations in general form:

If straight not perpendicular, then oriented the angle between them can be calculated using the formula:

Let's pay close attention to the denominator - this is exactly scalar product direction vectors of straight lines:

If , then the denominator of the formula vanishes, and the vectors will be orthogonal and the lines will be perpendicular. That is why a reservation was made about the non-perpendicularity of the lines in the formulation.

Based on the foregoing, the solution is conveniently formalized in two steps:

1) Calculate the scalar product of directing vectors of straight lines:
so the lines are not perpendicular.

2) We find the angle between the lines by the formula:

Using the inverse function, it is easy to find the angle itself. In this case, we use the oddness of the arc tangent (see Fig. Graphs and properties of elementary functions):

Answer:

In the answer, we indicate the exact value, as well as the approximate value (preferably both in degrees and in radians), calculated using a calculator.

Well, minus, so minus, it's okay. Here is a geometric illustration:

It is not surprising that the angle turned out to be of a negative orientation, because in the condition of the problem the first number is a straight line and the “twisting” of the angle began precisely from it.

If you really want to get a positive angle, you need to swap the straight lines, that is, take the coefficients from the second equation , and take the coefficients from the first equation . In short, you need to start with a direct .

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Slides captions:

Angle between lines

Aims and objectives of the lesson: To form the concept of the angle between: Intersecting; parallel; intersecting lines. Learn to find the angle between: Intersecting; parallel; intersecting lines.

Recall: The base of the prism ABCDA 1 B 1 C 1 D 1 is a trapezoid. Which of the following pairs of lines are crossing lines?

Location of lines in space and the angle between them 1. Intersecting lines. 2. Parallel lines. 3. Intersecting lines.

Any two intersecting lines lie in the same plane and form four non-expanded angles.

If intersecting lines form four equal angles, then the angle between these lines is 90°. a b

The angle between two parallel lines is 0°.

The angle between two intersecting lines in space is the smallest of the angles formed by the rays of these lines with the vertex at the point of their intersection.

The angle between intersecting lines a and b is the angle between the constructed intersecting lines and.

The angle between intersecting lines, as well as between lines of the same plane, cannot be more than 90 °. Two intersecting lines that form an angle of 90° are called perpendicular. a b a 1 c c 1 d

Angle between skew lines Let AB and CD be two skew lines. Let's take an arbitrary point M 1 of the space and draw lines A 1 B 1 and C 1 D 1 through it, respectively, parallel to lines AB and CD . A B C D A 1 B 1 C 1 D 1 M 1 φ If the angle between the lines A 1 B 1 and C 1 D 1 is equal to φ, then we will say that the angle between the intersecting lines AB and CD is equal to φ.

Find the angle between the skew lines AB and CD As a point M 1, you can take any point on one of the skew lines. A B C D M 1 A 1 B 1 φ

Physical education for the eyes

Show perpendicular intersecting lines in the environment.

Given an image of a cube. Find the angle between intersecting lines a and b. 90° 45° Answer Answer

Given an image of a cube. Find the angle between intersecting lines a and b. 90° 60° Answer Answer

Given an image of a cube. Find the angle between intersecting lines a and b 90° 90° Answer Answer

Homework: §4 (pp. 85-89), #268, #269.

Physical education minute

Problem №1 In a regular pyramid SABCD , all edges of which are equal to 1, the point E is the midpoint of the edge SC . Find the angle between lines AD and BE.

Class work: Tasks: No. 263 No. 265 No. 267

Preview:

APPROVE

Mathematic teacher

L. R. Volnyak

"__" ________ 2016

Subject : "Angle between lines"

Tutorials:

Developing:

Educational:

Lesson type: Learning new material.

Methods: verbal (story), visual (presentation), dialogic.

1. Organizing time.

• Greetings.

1. Knowledge update.

1. What is the relative position of two lines in space?
2. How many angles are formed when two lines intersect in space?
3. How to determine the angle between intersecting lines?

Slad3

1. Prism base ABCDA 1 B 1 C 1 D 1 - trapezoid. Which of the following pairs of lines are crossing lines?

Answer: AB and CC 1, A 1 D 1 and CC 1.

1. Learning new material.

slide 4

Location of lines in space and the angle between them.

1. Intersecting lines.
2. Parallel lines.
3. Crossing straight lines.

slide 5

Any two intersecting lines lie in the same plane and form four non-expanded angles.

slide 6

If intersecting lines form four equal angles, then the angle between these lines is 90°.

Slide 7

The angle between two parallel lines is 0°.

Slide 8

The angle between two intersecting lines in space is the smallest of the angles formed by the rays of these lines with the vertex at the point of their intersection.

Slide 9 a and b and .

Slide 10

The angle between intersecting lines, as well as between lines of the same plane, cannot be more than 90 °. Two intersecting lines that form an angle of 90° are called perpendicular.

slide 11

Angle between crossing lines.

Let AB and CD be two intersecting lines.

Take an arbitrary point M 1 space and draw straight lines A 1 in 1 and C 1 D 1 , respectively, parallel to lines AB and CD.

If the angle between the lines A 1 in 1 and C 1 D 1 is equal to φ, then we will say that the angle between the intersecting lines AB and CD is equal to φ.

slide 12

Find the angle between skew lines AB and CD.

As point M 1 one can take any point on one of the intersecting lines.

slide 13

Physical education minute

Slide 14

1. Show perpendicular intersecting lines in the environment.

slide 15

2. An image of a cube is given. Find the angle between intersecting lines a and b.

a) 90°; b) 45°;

slide 16

c) 60°; d) 90°;

Slide 17

e) 90°; f) 90°.

1. Fixing new material

Slide 19

Physical education minute

Slide 20

№1.

In the right pyramid SABCD , all edges of which are equal to 1, the point E - the middle of the rib SC .Find the angle between the lines AD and B.E.

Decision:

Desired angle = corner CBE .Triangle SBC is equilateral.

BE - angle bisector = 60. Angle CBE is 30.

Answer: 30°.

№263.

Answer:

Angle between skew lines a and b called the angle between the constructed intersecting lines a 1 and b 1 , and a 1 || a, b 1 || b.

№265.

The angle between straight lines a and b is 90°. Is it true that lines a and b intersect?

Answer:

False, since lines can either intersect or intersect.

№267.

DABC is a tetrahedron, point O and F are the midpoints of AD and CD, respectively, segment TK is the midline of triangle ABC.

1. What is the angle between lines OF and CB?
2. Is it true that the angle between lines OF and TK is 60°?
3. What is the angle between lines TF and DB?

Decision:

Given: DABC,

O is the middle of AD,

F is the middle of the CD,

TC is the middle line ∆ABC.

Decision:

1. Reflection

• What have we learned new?
• Did we cope with the tasks that were set at the beginning of the lesson?
• What problems have we learned to solve?

1. Homework.

§4 (pp. 85-89), #268, #269.

Preview:

APPROVE

Mathematic teacher

L. R. Volnyak

"__" ________ 2016

Subject : "Angle between lines"

Tutorials: with the help of practical tasks, to ensure that students understand the definition of the angle between intersecting, parallel and skew lines;

Developing: to develop the spatial imagination of students in solving geometric problems, geometric thinking, interest in the subject, cognitive and creative activity of students, mathematical speech, memory, attention; develop independence in the development of new knowledge.

Educational: to educate students in a responsible attitude to educational work, strong-willed qualities; to form an emotional culture and a culture of communication.

lesson type: generalization and systematization of knowledge and skills.

Methods: verbal (story), dialogical.

1. Organizing time.

• Greetings.
• Communication of the goals and objectives of the lesson.
• Motivation for learning new material.
• Psychological and pedagogical setting of students for the upcoming activities.
• Checking those present at the lesson;

1. Checking homework

№268

ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped, point O and T - the midpoints of the edges of the CC 1 and DD 1 respectively. a) Is it true that the angle between lines AD and TO is 90°? b) What is the angle between the lines A 1 B 1 and BC?

Decision:

a) True, because TO || DC =>(AD, TO) = ADC = 90° (ABCD is a rectangle).

b)BC || B 1 C 1 => (A 1 B 1 , BC) = A 1 B 1 C 1 = 90°.

Answer: 90°, 90°.

№269

ABCDA 1 B 1 C 1 D 1 - cube. a) Is it true that the angle between the lines A 1 B and C 1 D is 90°? b) Find the angle between the lines B 1 O and C 1 D. c) Is it true that the angle between lines AC and C 1D equals 45°?

Decision:

a) True, because B 1 A || C 1 D => (A 1 B, C 1 D)= (B 1 A, A 1 B) = 90°, as the angle between the diagonals of the square.

b) 1. B 1 A || C 1 D=> (B 1 O, C 1 D) = AB 1 O.

2. in Δ AB 1 C AB 1 \u003d B 1 C = AC as diagonals of equal squares B 1 O - median and bisector AB 1 C=60° => AB 1 O=30°.

c) no, since C 1 D || BA => (AC, C 1 D) \u003d B 1 AC=60° as an equilateral angle Δ AB 1 C.

Answer: b) 30°.

1. Knowledge update.

Method: frontal survey (oral):

1. What branches does geometry study?
2. What is the angle between parallel lines?
3. What figures are studied by planimetry, and which are solid geometry?
4. What is the skew angle?
5. What are two intersecting lines that form an angle of 90° called?

1. Consolidation of what has been learned.

Dictation (10 min):

Option 1:

The edge of the cube is a .

Find: (AB 1 ,SS 1 )

Decision:

SS1‖BB1

(AB1,CC1) = AB1B

AB1B=45˚

Answer: (AB1, SS1) = 45˚

1. Let a and b be intersecting lines, and the line b 1 || b. Is it true that the angle between lines a and b is equal to the angle between lines a and b 1 ? If yes, why?

Option 2:

1. What is the angle between skew lines?

The edge of the cube is a .

In this article, we will first define the angle between skew lines and give a graphic illustration. Next, we answer the question: "How to find the angle between skew lines if the coordinates of the direction vectors of these lines in a rectangular coordinate system are known"? In conclusion, we will practice finding the angle between skew lines when solving examples and problems.

Page navigation.

Angle between skew lines - definition.

We will gradually approach the definition of the angle between intersecting lines.

Let us first recall the definition of skew lines: two lines in three-dimensional space are called interbreeding if they do not lie in the same plane. It follows from this definition that skew lines do not intersect, are not parallel, and, moreover, do not coincide, otherwise they would both lie in some plane.

We present some additional auxiliary arguments.

Let two intersecting lines a and b be given in three-dimensional space. Let us construct the lines a 1 and b 1 so that they are parallel to the skew lines a and b, respectively, and pass through some point in the space M 1 . Thus, we will get two intersecting lines a 1 and b 1 . Let the angle between the intersecting lines a 1 and b 1 be equal to the angle . Now let's construct lines a 2 and b 2 , parallel to skew lines a and b, respectively, passing through the point M 2 , which is different from the point M 1 . The angle between the intersecting lines a 2 and b 2 will also be equal to the angle. This statement is true, since the lines a 1 and b 1 will coincide with the lines a 2 and b 2, respectively, if you perform a parallel transfer, in which the point M 1 goes to the point M 2. Thus, the measure of the angle between two lines intersecting at the point M, respectively parallel to the given skew lines, does not depend on the choice of the point M.

We are now ready to define the angle between skew lines.

Definition.

Angle between skew lines is the angle between two intersecting lines that are respectively parallel to the given skew lines.

It follows from the definition that the angle between the skew lines will also not depend on the choice of the point M . Therefore, as a point M, you can take any point belonging to one of the skew lines.

We give an illustration of the definition of the angle between skew lines.

Finding the angle between skew lines.

Since the angle between intersecting lines is determined by the angle between intersecting lines, finding the angle between intersecting lines is reduced to finding the angle between the corresponding intersecting lines in three-dimensional space.

Undoubtedly, the methods studied in geometry lessons in high school are suitable for finding the angle between skew lines. That is, having completed the necessary constructions, it is possible to connect the desired angle with any angle known from the condition, based on the equality or similarity of the figures, in some cases it will help cosine theorem, and sometimes leads to the result definition of sine, cosine and tangent of an angle right triangle.

However, it is very convenient to solve the problem of finding the angle between skew lines using the coordinate method. That is what we will consider.

Let Oxyz be introduced in three-dimensional space (however, in many problems it has to be introduced independently).

Let's set ourselves the task: to find the angle between the intersecting lines a and b, which correspond to some equations of the line in space in the rectangular coordinate system Oxyz.

Let's solve it.

Let's take an arbitrary point of the three-dimensional space M and assume that the lines a 1 and b 1 pass through it, parallel to the intersecting lines a and b, respectively. Then the required angle between intersecting lines a and b is equal to the angle between intersecting lines a 1 and b 1 by definition.

Thus, it remains for us to find the angle between the intersecting lines a 1 and b 1 . To apply the formula for finding the angle between two intersecting lines in space, we need to know the coordinates of the direction vectors of the lines a 1 and b 1 .

How can we get them? And it's very simple. The definition of the directing vector of a straight line allows us to state that the sets of directing vectors of parallel straight lines coincide. Therefore, as the direction vectors of the lines a 1 and b 1, we can take the direction vectors and straight lines a and b, respectively.

So, the angle between two intersecting lines a and b is calculated by the formula
, where and are the direction vectors of the lines a and b, respectively.

Formula for finding the cosine of the angle between skew lines a and b has the form .

Allows you to find the sine of the angle between skew lines if the cosine is known: .

It remains to analyze the solutions of the examples.

Example.

Find the angle between the skew lines a and b , which are defined in the Oxyz rectangular coordinate system by the equations and .

Decision.

The canonical equations of a straight line in space allow you to immediately determine the coordinates of the directing vector of this straight line - they are given by numbers in the denominators of fractions, that is, . Parametric equations of a straight line in space also make it possible to immediately write down the coordinates of the direction vector - they are equal to the coefficients in front of the parameter, that is, - direction vector straight . Thus, we have all the necessary data to apply the formula by which the angle between skew lines is calculated:

Answer:

The angle between the given skew lines is .

Example.

Find the sine and cosine of the angle between the skew lines on which the edges AD and BC of the pyramid ABCD lie, if the coordinates of its vertices are known:.

Decision.

The direction vectors of the crossing lines AD and BC are the vectors and . Let's calculate their coordinates as the difference between the corresponding coordinates of the end and start points of the vector:

According to the formula we can calculate the cosine of the angle between the given skew lines:

Now we calculate the sine of the angle between the skew lines: