What is the angle between a straight line and a plane. The angle between a straight line and a plane. Perpendicularity of a line and a plane

The article begins with the definition of the angle between a straight line and a plane. This article will show you how to find the angle between a straight line and a plane using the coordinate method. The solutions to examples and problems will be discussed in detail.

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First, it is necessary to repeat the concept of a straight line in space and the concept of a plane. To determine the angle between a straight line and a plane, several auxiliary definitions. Let's look at these definitions in detail.

Definition 1

A straight line and a plane intersect in the case when they have one common point, that is, it is the intersection point of a straight line and a plane.

A straight line intersecting a plane may be perpendicular to the plane.

Definition 2

A straight line is perpendicular to a plane when it is perpendicular to any line located in this plane.

Definition 3

Projection of point M onto a planeγ is the point itself if it lies in given plane, or is the point of intersection of the plane with a line perpendicular to the plane γ passing through the point M, provided that it does not belong to the plane γ.

Definition 4

Projection of line a onto a planeγ is the set of projections of all points of a given line onto the plane.

From this we obtain that the projection of a line perpendicular to the plane γ has an intersection point. We find that the projection of line a is a line belonging to the plane γ and passing through the intersection point of line a and the plane. Let's look at the figure below.

On at the moment we have all the necessary information and data to formulate the definition of the angle between a straight line and a plane

Definition 5

The angle between a straight line and a plane the angle between this straight line and its projection onto this plane is called, and the straight line is not perpendicular to it.

The definition of angle given above helps to come to the conclusion that the angle between a line and a plane is the angle between two intersecting lines, that is, a given line along with its projection onto the plane. This means that the angle between them will always be acute. Let's take a look at the picture below.

The angle located between a straight line and a plane is considered to be right, that is, equal to 90 degrees, but the angle located between parallel straight lines is not defined. There are cases when its value is taken equal to zero.

Problems where it is necessary to find the angle between a straight line and a plane have many variations in solution. The course of the solution itself depends on the available data on the condition. Frequent companions to the solution are signs of similarity or equality of figures, cosines, sines, tangents of angles. Finding the angle is possible using the coordinate method. Let's look at it in more detail.

If a rectangular coordinate system O x y z is introduced in three-dimensional space, then a straight line a is specified in it, intersecting the plane γ at point M, and it is not perpendicular to the plane. It is necessary to find the angle α located between a given straight line and the plane.

First you need to apply the definition of the angle between a straight line and a plane using the coordinate method. Then we get the following.

In the coordinate system O x y z, a straight line a is specified, which corresponds to the equations of the straight line in space and the directing vector of the straight line in space; for the plane γ there corresponds the equation of the plane and the normal vector of the plane. Then a → = (a x , a y , a z) is the direction vector of the given line a, and n → (n x , n y , n z) is the normal vector for the plane γ. If we imagine that we have the coordinates of the direction vector of the straight line a and the normal vector of the plane γ, then their equations are known, that is, they are specified by condition, then it is possible to determine the vectors a → and n → based on the equation.

To calculate the angle, it is necessary to transform the formula to obtain the value of this angle using the existing coordinates of the directing vector of the straight line and the normal vector.

It is necessary to plot the vectors a → and n →, starting from the point of intersection of the straight line a with the plane γ. There are 4 options for the location of these vectors relative to given lines and planes. Look at the picture below, which shows all 4 variations.

From here we obtain that the angle between the vectors a → and n → is designated a → , n → ^ and is acute, then the desired angle α located between the straight line and the plane is complemented, that is, we obtain an expression of the form a → , n → ^ = 90 ° - α. When, by condition, a →, n → ^ > 90 °, then we have a →, n → ^ = 90 ° + α.

From here we have that the cosines of equal angles are equal, then the last equalities are written in the form of a system

cos a → , n → ^ = cos 90 ° - α , a → , n → ^< 90 ° cos a → , n → ^ = cos 90 ° + α , a → , n → ^ >90°

You must use reduction formulas to simplify expressions. Then we obtain equalities of the form cos a → , n → ^ = sin α , a → , n → ^< 90 ° cos a → , n → ^ = - s i n α , a → , n → ^ >90°

After carrying out the transformations, the system takes the form sin α = cos a → , n → ^ , a → , n → ^< 90 ° sin α = - cos a → , n → ^ , a → , n → ^ >90 ° ⇔ sin α = cos a → , n → ^ , a → , n → ^ > 0 sin α = - cos a → , n → ^ , a → , n → ^< 0 ⇔ ⇔ sin α = cos a → , n → ^

From this we obtain that the sine of the angle between the straight line and the plane is equal to the modulus of the cosine of the angle between the directing vector of the straight line and the normal vector of the given plane.

The section on finding the angle formed by two vectors revealed that this angle takes the value dot product vectors and the product of these lengths. The process of calculating the sine of the angle obtained by the intersection of a straight line and a plane is performed according to the formula

sin α = cos a → , n → ^ = a → , n → ^ a → n → = a x n x + a y n y + a z n z a x 2 + a y 2 + a z 2 n x 2 + n y 2 + n z 2

This means that the formula for calculating the angle between a straight line and a plane with the coordinates of the directing vector of the straight line and the normal vector of the plane after transformation is of the form

α = a r c sin a → , n → ^ a → n → = a r c sin a x n x + a y n y + a z n z a x 2 + a y 2 + a z 2 n x 2 + n y 2 + n z 2

Finding the cosine for a known sine is possible by applying the basic trigonometric identity. The intersection of a straight line and a plane forms an acute angle. This suggests that its value will be a positive number, and its calculation is made from the formula cos α = 1 - sin α.

Let's solve several similar examples to consolidate the material.

Example 1

Find the angle, sine, cosine of the angle formed by the straight line x 3 = y + 1 - 2 = z - 11 6 and the plane 2 x + z - 1 = 0.

Solution

To obtain the coordinates of the direction vector, it is necessary to consider the canonical equations of a straight line in space. Then we get that a → = (3, - 2, 6) is the direction vector of the straight line x 3 = y + 1 - 2 = z - 11 6.

To find the coordinates of a normal vector, it is necessary to consider the general equation of the plane, since their presence is determined by the coefficients available in front of variables of the equation. Then we find that for the plane 2 x + z - 1 = 0 the normal vector has the form n → = (2, 0, 1).

It is necessary to proceed to calculating the sine of the angle between the straight line and the plane. To do this, it is necessary to substitute the coordinates of the vectors a → and b → into the given formula. We get an expression of the form

sin α = cos a → , n → ^ = a → , n → ^ a → n → = a x n x + a y n y + a z n z a x 2 + a y 2 + a z 2 n x 2 + n y 2 + n z 2 = = 3 2 + (- 2) 0 + 6 1 3 2 + (- 2) 2 + 6 2 2 2 + 0 2 + 1 2 = 12 7 5

From here we find the value of the cosine and the value of the angle itself. We get:

cos α = 1 - sin α = 1 - 12 7 5 2 = 101 7 5

Answer: sin α = 12 7 5, cos α = 101 7 5, α = a r c cos 101 7 5 = a r c sin 12 7 5.

Example 2

There is a pyramid built using the values of the vectors A B → = 1, 0, 2, A C → = (- 1, 3, 0), A D → = 4, 1, 1. Find the angle between straight line A D and plane A B C.

Solution

To calculate the desired angle, it is necessary to have the coordinates of the directing vector of the straight line and the normal vector of the plane. for a straight line A D the direction vector has coordinates A D → = 4, 1, 1.

The normal vector n → belonging to the plane A B C is perpendicular to the vector A B → and A C →. This implies that the normal vector of the plane A B C can be considered vector product vectors A B → and A C → . We calculate this using the formula and get:

n → = A B → × A C → = i → j → k → 1 0 2 - 1 3 0 = - 6 · i → - 2 · j → + 3 · k → ⇔ n → = (- 6 , - 2 , 3 )

It is necessary to substitute the coordinates of the vectors to calculate the desired angle formed by the intersection of a straight line and a plane. we get an expression of the form:

α = a r c sin A D → , n → ^ A D → · n → = a r c sin 4 · - 6 + 1 · - 2 + 1 · 3 4 2 + 1 2 + 1 2 · - 6 2 + - 2 2 + 3 2 = a r c sin 23 21 2

Answer: a r c sin 23 21 2 .

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The definition of the angle between a straight line and a plane is based on the concept of oblique projection. Definition. The angle between a straight line and a plane is the angle between this straight line and its projection onto a given plane.

In Fig. 341 shows the angle a between the inclined AM and its projection onto the plane K.

Note. If a straight line is parallel to a plane or lies in it, then its angle with the plane is considered equal to zero. If it is perpendicular to the plane, then the angle is declared to be right (the previous definition is literally inapplicable here!). In other cases, an acute angle is implied between the straight line and its projection. Therefore, the angle between a straight line and a plane never exceeds a right angle. Let us also note that here it is more correct to talk about the measure of an angle, and not about an angle (indeed, we are talking about the measure of the inclination of a straight line to a plane, but the concept of an angle as a flat figure bounded by two rays has no direct relation here).

Let us verify one more property of an acute angle between a straight line and a plane.

Of all the angles formed by a given straight line and all possible straight lines in a plane, the angle with the projection of a given straight line is the smallest.

Proof. Let's turn to Fig. 342. Let a be a given line, its projection onto the plane be an arbitrary other line in the plane K (for convenience, we drew it through the point A of the intersection of line a with the plane). Let us put it on a straight segment, i.e., equal to the base of the inclined MA, where is the projection of one of the points of the inclined a.

Then in triangles two sides are equal: side AM is common, they are equal in construction. But the third side in the triangle is greater than the third side in the triangle (the inclined side is greater than the perpendicular). This means that the opposite angle b is greater than the corresponding angle a b (see paragraph 217): , which is what needed to be proven.

The angle between a straight line and a plane is the smallest of the angles between a given straight line and all possible straight lines in the plane.

Fair and so

Theorem. Acute angle between a straight line lying in a plane and the projection of an inclined one onto this plane is less than the angle between this straight line and the inclined one.

Proof. Let be a straight line lying in the plane (Fig. 342), a be inclined to the plane, t be its projection onto the plane. We will consider the straight line as inclined to the plane, then it will be its projection onto the indicated plane and, using the previous property, we will find: which is what we needed to prove. From the theorem of three perpendiculars it is clear that in the case when a straight line in a plane is perpendicular to, the projection of an oblique one (the case is not acute, but right angle), the straight line is also perpendicular to the most inclined one; in this case, both angles we are talking about are right angles and therefore equal to each other.

Let some rectangular coordinate system and a straight line be given . Let And - two different planes intersecting in a straight line and given accordingly by equations. These two equations jointly define the straight line if and only if they are not parallel and do not coincide with each other, i.e. normal vectors
And
these planes are not collinear.

Definition. If the coefficients of the equations

are not proportional, then these equations are called general equations straight line, defined as the line of intersection of planes.

Definition. Any non-zero vector parallel to a line is called guide vector this straight line.

Let us derive the equation of the straight line passing through a given point
space and having a given direction vector
.

Let the point
- arbitrary point on a straight line . This point lies on a line if and only if the vector
, having coordinates
, collinear to the direction vector
direct. According to (2.28), the condition for collinearity of vectors
And looks like

. (3.18)

Equations (3.18) are called canonical equations straight line passing through a point
and having a direction vector
.

If straight is given by general equations (3.17), then the direction vector this line is orthogonal to the normal vectors
And
planes specified by equations. Vector
according to the vector product property, it is orthogonal to each of the vectors And . According to the definition, as a direction vector direct you can take a vector
, i.e.
.

To find a point
consider the system of equations
. Since the planes defined by the equations are not parallel and do not coincide, then at least one of the equalities does not hold
. This leads to the fact that at least one of the determinants ,
,
different from zero. For definiteness, we will assume that
. Then, taking an arbitrary value , we obtain a system of equations for the unknowns And :

According to Cramer's theorem, this system has a unique solution defined by the formulas

,
. (3.19)

If you take
, then the straight line given by equations (3.17) passes through the point
.

Thus, for the case when
, the canonical equations of the line (3.17) have the form

The canonical equations of the straight line (3.17) are written similarly for the case when the determinant is nonzero
or
.

If a line passes through two different points
And
, then its canonical equations have the form

. (3.20)

This follows from the fact that the straight line passes through the point
and has a direction vector.

Let us consider the canonical equations (3.18) of the straight line. Let us take each of the relations as a parameter , i.e.
. One of the denominators of these fractions is non-zero, and the corresponding numerator can take any value, so the parameter can take on any real values. Considering that each of the ratios is equal , we get parametric equations direct:

,
,
. (3.21)

Let the plane is given by a general equation, and the straight line - parametric equations
,
,
. Dot
intersection of a straight line and planes must simultaneously belong to a plane and a line. This is only possible if the parameter satisfies the equation, i.e.
. Thus, the point of intersection of a straight line and a plane has coordinates

Example 32. Write parametric equations for a line passing through the points
And
.

Solution. For the directing vector of the straight line we take the vector

. A straight line passes through a point , therefore, according to formula (3.21), the required straight line equations have the form
,
,
.

Example 33. Vertices of the triangle
have coordinates
,
And
respectively. Compose parametric equations for the median drawn from the vertex .

Solution. Let
- middle of the side
, Then
,
,
. As the guide vector of the median, we take the vector
. Then the parametric equations of the median have the form
,
,
.

Example 34. Compose the canonical equations of a line passing through a point
parallel to the line
.

Solution. The straight line is defined as the line of intersection of planes with normal vectors
And
. As a guide vector take the vector of this line
, i.e.
. According to (3.18), the required equation has the form
or
.

3.8. The angle between straight lines in space. Angle between a straight line and a plane

Let two straight lines And in space are given by their canonical equations
And
. Then one of the corners between these lines is equal to the angle between their direction vectors
And
. Using formula (2.22), to determine the angle we get the formula

. (3.22)

Second corner between these lines is equal
And
.

Condition for parallel lines And is equivalent to the condition of collinearity of vectors
And
and lies in the proportionality of their coordinates, i.e. the condition for parallel lines has the form

. (3.23)

If straight And are perpendicular, then their direction vectors are orthogonal, i.e. the perpendicularity condition is determined by the equality

. (3.24)

Consider a plane , given by the general equation, and the straight line , given by the canonical equations
.

Corner between the straight line and plane is complementary to the angle between the directing vector of the straight line and the normal vector of the plane, i.e.
And
, or

. (3.24)

Condition for parallelism of a line and planes is equivalent to the condition that the direction vector of the line and the normal vector of the plane are perpendicular, i.e. the scalar product of these vectors must be equal to zero:

If the line is perpendicular to the plane, then the direction vector of the line and the normal vector of the plane must be collinear. In this case, the coordinates of the vectors are proportional, i.e.

. (3.26)

Example 35. Find obtuse angle between straight lines
,
,
And
,
,
.

Solution. The direction vectors of these lines have coordinates
And
. Therefore one corner between straight lines is determined by the ratio, i.e.
. Therefore, the condition of the problem is satisfied by the second angle between the lines, equal to
.

3.9. Distance from a point to a line in space

Let
 point in space with coordinates
, straight line given by canonical equations
. Let's find the distance from point
to a straight line .

Let's apply a guide vector
to the point
. Distance from point
to a straight line is the height of a parallelogram built on vectors And
. Let's find the area of a parallelogram using the cross product:

On the other side, . From the equality of the right-hand sides of the last two relations it follows that

. (3.27)

3.10. Ellipsoid

Definition. Ellipsoid is a second-order surface, which in some coordinate system is defined by the equation

. (3.28)

Equation (3.28) is called the canonical equation of the ellipsoid.

From equation (3.28) it follows that the coordinate planes are planes of symmetry of the ellipsoid, and the origin of coordinates is the center of symmetry. Numbers
are called semi-axes of the ellipsoid and represent the lengths of segments from the origin to the intersection of the ellipsoid with the coordinate axes. An ellipsoid is a bounded surface enclosed in a parallelepiped
,
,
.

Let us establish the geometric form of the ellipsoid. To do this, let us find out the shape of the lines of intersection of its planes parallel to the coordinate axes.

To be specific, consider the lines of intersection of the ellipsoid with the planes
, parallel to the plane
. Equation for the projection of the intersection line onto a plane
is obtained from (3.28) if we put in it
. The equation of this projection is

. (3.29)

If
, then (3.29) is the equation of an imaginary ellipse and the points of intersection of the ellipsoid with the plane
No. It follows from this that
. If
, then line (3.29) degenerates into points, i.e. planes
touch the ellipsoid at points
And
. If
, That
and you can introduce the notation

,
. (3.30)

Then equation (3.29) takes the form

, (3.31)

i.e. projection onto a plane
lines of intersection of the ellipsoid and the plane
is an ellipse with semi-axes, which are determined by equalities (3.30). Since the line of intersection of the surface with planes parallel to the coordinate planes is a projection “raised” to a height , then the intersection line itself is an ellipse.

When decreasing the value axle shafts And increase and reach their greatest value at
, i.e. in the section of the ellipsoid by the coordinate plane
the largest ellipse with semi-axes is obtained
And
.

The idea of an ellipsoid can be obtained in another way. Consider on the plane
family of ellipses (3.31) with semi-axes And , defined by relations (3.30) and depending on . Each such ellipse is a level line, that is, a line at each point of which the value the same. “Raising” each such ellipse to a height , we obtain a spatial view of the ellipsoid.

A similar picture is obtained when a given surface is intersected by planes parallel to the coordinate planes
And
.

Thus, an ellipsoid is a closed elliptical surface. In case
The ellipsoid is a sphere.

The line of intersection of an ellipsoid with any plane is an ellipse, since such a line is a limited line of the second order, and the only limited line of the second order is an ellipse.

This means finding the angle between this line and its projection onto a given plane.

A spatial model illustrating the task is presented in the figure.

Problem solution plan:
1. From an arbitrary point A∈a lower the perpendicular to the plane α ;
2. Determine the meeting point of this perpendicular with the plane α . Dot A α- orthogonal projection A to the plane α ;
3. Find the point of intersection of the line a with plane α . Dot a α- straight trail a on the plane α ;
4. We carry out ( A α a α) - projection of a straight line a to the plane α ;
5. Determine the real value ∠ Aa α A α, i.e. ∠ φ .

Problem solution find the angle between a line and a plane can be greatly simplified if we do not define ∠ φ between a straight line and a plane, and complementary to 90° ∠ γ . In this case, there is no need to determine the projection of the point A and straight line projections a to the plane α . Knowing the magnitude γ , calculated by the formula:

$ φ = 90° - γ $

a and plane α , defined by parallel lines m And n.

a α
By rotating around the horizontal line specified by points 5 and 6, we determine the natural size ∠ γ . Knowing the magnitude γ , calculated by the formula:

$ φ = 90° - γ $

Determining the angle between a straight line a and plane α , defined by triangle BCD.

From an arbitrary point on a line a lower the perpendicular to the plane α
By rotating around the horizontal line specified by points 3 and 4, we determine the natural size ∠ γ . Knowing the magnitude γ , calculated using the formula.