Corner between straight lines in space we will call any of the adjacent angles formed by two straight lines drawn through an arbitrary point parallel to the data.
Let two straight lines be given in space:
Obviously, the angle between the straight lines can be taken as the angle between their direction vectors and. Since, then, according to the formula for the cosine of the angle between the vectors, we get
The conditions for parallelism and perpendicularity of two straight lines are equivalent to the conditions for parallelism and perpendicularity of their direction vectors and:
Two straight parallel if and only if their respective coefficients are proportional, i.e. l 1 parallel l 2 if and only if parallel 
.
Two straight perpendicular if and only if the sum of the products of the corresponding coefficients is zero:.
Have goal between straight line and plane
Let it be straight d- not perpendicular to the plane θ;
d′ - projection of the straight line d on the plane θ;
The smallest of the angles between straight lines d and d′ We will call angle between line and plane.
We denote it as φ = ( d,θ)
If d⊥θ, then ( d, θ) = π / 2
Oi→j→k→ - rectangular coordinate system.
Plane equation:
θ: Ax+By+Cz+D=0
We assume that the line is given by a point and a direction vector: d[M 0,p→]
Vector n→(A,B,C)⊥θ
Then it remains to find out the angle between the vectors n→ and p→, we denote it as γ = ( n→,p→).
If the angle γ<π/2 , то искомый угол φ=π/2−γ .
If the angle γ> π / 2, then the sought angle φ = γ − π / 2
sinφ = sin (2π − γ) = cosγ
sinφ = sin (γ − 2π) = - cosγ
Then, angle between line and plane can be calculated using the formula:
sinφ = ∣cosγ∣ = ∣ ∣ Ap 1+Bp 2+Cp 3∣ ∣ √A 2+B 2+C 2√p 21+p 22+p 23
Question29. The concept of a quadratic form. Sign-definiteness of quadratic forms.
Quadratic form j (x 1, x 2, ..., x n) n real variables x 1, x 2, ..., x n called the sum of the form 
, (1)
where a ij - some numbers called coefficients. Without loss of generality, we can assume that a ij = a ji.
The quadratic form is called valid, if a ij
Î GR. By a matrix of quadratic form called a matrix composed of its coefficients. The quadratic form (1) corresponds to the only symmetric matrix 
Ie. A T = A... Therefore, the quadratic form (1) can be written in matrix form j ( NS) = x T Ax, where x T = (NS 1 NS 2 … x n). (2)
And, conversely, every symmetric matrix (2) corresponds to a unique quadratic form up to the notation of the variables.
By the rank of the quadratic form call the rank of its matrix. The quadratic form is called non-degenerate, if its matrix is nondegenerate A... (recall that the matrix A is called nondegenerate if its determinant is not zero). Otherwise, the quadratic form is degenerate.
positively defined(or strictly positive) if
j ( NS) > 0 , for anyone NS = (NS 1 , NS 2 , …, x n), except NS = (0, 0, …, 0).
Matrix A positive definite quadratic form j ( NS) is also called positive definite. Consequently, a single positive definite matrix corresponds to a positive definite quadratic form and vice versa.
The quadratic form (1) is called negatively defined(or strictly negative) if
j ( NS) < 0, для любого NS = (NS 1 , NS 2 , …, x n), except NS = (0, 0, …, 0).
Similarly as above, a matrix of negative definite quadratic form is also called negative definite.
Therefore, the positively (negatively) definite quadratic form j ( NS) reaches the minimum (maximum) value j ( NS*) = 0 for NS* = (0, 0, …, 0).
Note that most of the quadratic forms are not definite, that is, they are neither positive nor negative. Such quadratic forms vanish not only at the origin of the coordinate system, but also at other points.
When n> 2, special criteria are required to check the definiteness of the quadratic form. Let's consider them.
Major Minors the quadratic form are called minors:
that is, these are minors of order 1, 2, ..., n matrices A located in the upper left corner, the last of them coincides with the determinant of the matrix A.
Positive definiteness criterion (Sylvester criterion)
NS) = x T Ax was positive definite, it is necessary and sufficient that all the principal minors of the matrix A were positive, that is: M 1 > 0, M 2 > 0, …, M n > 0. Negative certainty criterion In order for the quadratic form j ( NS) = x T Ax was negative definite, it is necessary and sufficient that its principal minors of even order are positive, and that of odd order are negative, i.e.: M 1 < 0, M 2 > 0, M 3 < 0, …, (–1)n
With this online calculator you can find the angle between straight lines. A detailed solution with explanations is given. To calculate the angle between straight lines, set the dimension (2 - if a straight line on a plane is considered, 3 - if a straight line in space is considered), enter the elements of the equation into the cells and click on the "Solve" button. See the theoretical part below.
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Data entry instructions. Numbers are entered as whole numbers (examples: 487, 5, -7623, etc.), decimal numbers (eg 67., 102.54, etc.) or fractions. The fraction must be typed in the form a / b, where a and b (b> 0) are integers or decimal numbers. Examples 45/5, 6.6 / 76.4, -7 / 6.7, etc.
1. The angle between straight lines on the plane
The straight lines are given by the canonical equations
1.1. Determination of the angle between straight lines
Let in two-dimensional space the lines L 1 and L
Thus, from formula (1.4), one can find the angle between the straight lines L 1 and L 2. As can be seen from Fig. 1, intersecting straight lines form adjacent corners φ and φ 1 . If the found angle is greater than 90 °, then you can find the minimum angle between the straight lines L 1 and L 2: φ 1 =180-φ .
From formula (1.4), one can derive the conditions for parallelism and perpendicularity of two straight lines.
Example 1. Determine the angle between straight lines
Let's simplify and solve:
1.2. Parallelism condition for straight lines
Let be φ = 0. Then cosφ= 1. In this case, expression (1.4) will take the following form:
| , |
| , |
Example 2. Determine if lines are parallel
Equality (1.9) is satisfied, hence the lines (1.10) and (1.11) are parallel.
Answer. Lines (1.10) and (1.11) are parallel.
1.3. Condition of perpendicularity of straight lines
Let be φ = 90 °. Then cosφ= 0. In this case, expression (1.4) will take the following form:
Example 3. Determine if lines are perpendicular
Condition (1.13) is satisfied, hence straight lines (1.14) and (1.15) are perpendicular.
Answer. Lines (1.14) and (1.15) are perpendicular.
The straight lines are given by the general equations
1.4. Determination of the angle between straight lines
Let two lines L 1 and L 2 are given by the general equations
From the definition of the dot product of two vectors, we have:
Example 4. Find the angle between straight lines
Substituting the values A 1 , B 1 , A 2 , B 2 in (1.23), we get:
This angle is greater than 90 °. Find the minimum angle between the straight lines. To do this, subtract this angle from 180:
On the other hand, the parallelism condition for straight lines L 1 and L 2 is equivalent to the condition of collinear vectors n 1 and n 2 and can be represented as follows:
Equality (1.24) is satisfied, hence the lines (1.26) and (1.27) are parallel.
Answer. Lines (1.26) and (1.27) are parallel.
1.6. Condition of perpendicularity of straight lines
Condition of perpendicularity of straight lines L 1 and L 2 can be extracted from formula (1.20) by substituting cos(φ ) = 0. Then the scalar product ( n 1 ,n 2) = 0. Where
Equality (1.28) is satisfied, hence straight lines (1.29) and (1.30) are perpendicular.
Answer. Straight lines (1.29) and (1.30) are perpendicular.
2. The angle between straight lines in space
2.1. Determination of the angle between straight lines
Let the lines in space L 1 and L 2 are given by the canonical equations
where | q 1 | and | q 2 | direction vector modules q 1 and q 2 respectively, φ is the angle between vectors q 1 and q 2 .
From expression (2.3) we obtain:
 .
|
Let's simplify and solve:
 .
|
Find the corner φ
Each student who is preparing for the exam in mathematics will find it useful to repeat the topic "Finding the angle between straight lines." As statistics show, when passing the certification test, problems in this section of stereometry cause difficulties for a large number of students. At the same time, tasks that require finding the angle between straight lines are found in the USE of both the basic and the profile level. This means that everyone should be able to solve them.
Basic moments
There are 4 types of mutual arrangement of straight lines in space. They can coincide, intersect, be parallel, or intersect. The angle between them can be sharp or right.
To find the angle between the straight lines in the exam 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 using classic 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 do all the calculations correctly. The educational project "Shkolkovo" will help you to hone your skills in solving problems in stereometry and other sections of the school course.
Instructions
note
The period of the trigonometric function of the tangent is 180 degrees, which means that the slopes of the straight lines cannot, in absolute value, exceed this value.
Helpful advice
If the slopes are equal to each other, then the angle between such lines is 0, since such lines either coincide or are parallel.
To determine the value of the angle between crossing straight lines, it is necessary to move both straight lines (or one of them) to a new position using the parallel transfer method before crossing. After that, you should find the value of the angle between the resulting intersecting straight lines.
You will need
- Ruler, right triangle, pencil, protractor.
Instructions
So, let a vector V = (a, b, c) and a plane A x + B y + C z = 0 be given, where A, B and C are the coordinates of the normal N. Then the cosine of the angle α between vectors V and N is equal to: сos α = (a A + b B + c C) / (√ (a² + b² + c²) √ (A² + B² + C²)).
To calculate the value of the angle in degrees or radians, you need to calculate the function inverse to the cosine from the resulting expression, i.e. inverse cosine: α = arssos ((a A + b B + c C) / (√ (a² + b² + c²) √ (A² + B² + C²))).
Example: find injection between vector(5, -3, 8) and plane given by the general equation 2 x - 5 y + 3 z = 0 Solution: write down the coordinates of the normal vector of the plane N = (2, -5, 3). Substitute all known values into the above formula: cos α = (10 + 15 + 24) / √3724 ≈ 0.8 → α = 36.87 °.
Related Videos
A straight line that has one point in common with a circle is tangent to the circle. Another feature of the tangent is that it is always perpendicular to the radius drawn to the tangent point, that is, the tangent and the radius form a straight line injection... If from one point A two tangents are drawn to the circle AB and AC, then they are always equal to each other. Determining the angle between tangents ( injection ABC) is produced using the Pythagorean theorem.
Instructions
To determine the angle, you need to know the radius of the circle OB and OS and the distance of the start point of the tangent from the center of the circle - O. So, the angles ABO and ASO are equal, the radius of OB, for example, 10 cm, and the distance to the center of the circle AO is 15 cm. Determine the length of the tangent along the formula in accordance with the Pythagorean theorem: AB = square root of AO2 - OB2 or 152 - 102 = 225 - 100 = 125;
If on a straight line in space we mark two arbitrary points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2), then the coordinates of these points must satisfy the equation of the straight line obtained above:
In addition, for point M 1 you can write:
.
Solving these equations together, we get:
.
This is the equation of a straight line passing through two points in space.
General equations of a straight line in space.
The equation of a straight line can be considered as the equation of the line of intersection of two planes.
General equations of a straight line in coordinate form:
A practical task often consists in bringing the equations of straight lines in general form to the canonical form.
To do this, you need to find an arbitrary point of the line and the numbers m, n, p.
In this case, the directing vector of the straight line can be found as the cross product of the vectors of the normal to the given planes.
Example. Find the canonical equation if the straight line is given in the form:
To find an arbitrary point on a straight line, we take its coordinate x = 0, and then substitute this value into the given system of equations.
Those. A (0, 2, 1).
Find the components of the directing vector of the straight line.
Then the canonical equations of the straight line:
Example. Bring the equation of a straight line to canonical form, given in the form:
To find an arbitrary point of a straight line that is the line of intersection of the above planes, we take z = 0. Then:
;
2x - 9x - 7 = 0;
We get: A (-1; 3; 0).
Direction vector of a straight line: 
.
The angle between the planes.
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The angle between two planes in space is related to the angle between the normals to these planes 1 by the ratio: = 1 or = 180 0 - 1, i.e.
cos = cos 1.
Let's define the angle 1. It is known that planes can be specified by the ratios:
, where
(A 1, B 1, C 1), (A 2, B 2, C 2). We find the angle between the vectors of the normal from their dot product:
.
Thus, the angle between the planes is found by the formula:
The choice of the cosine sign depends on which angle between the planes should be found - acute, or adjacent to it obtuse.
Conditions for parallelism and perpendicularity of planes.
Based on the formula obtained above for finding the angle between the planes, it is possible to find the conditions for parallelism and perpendicularity of the planes.
In order for the planes to be perpendicular, it is necessary and sufficient that the cosine of the angle between the planes is equal to zero. This condition is met if:
The planes are parallel, the normal vectors are collinear: . This condition is satisfied if: 
.
The angle between straight lines in space.
Let two straight lines be given in space. Their parametric equations:
The angle between the straight lines and the angle between the direction vectors of these straight lines are related by the ratio: = 1 or = 180 0 - 1. The angle between the direction vectors is found from the dot product. Thus:
.
Conditions for parallelism and perpendicularity of straight lines in space.
For two straight lines to be parallel, it is necessary and sufficient that the direction vectors of these straight lines are collinear, i.e. their respective coordinates were proportional.