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Angle between a straight line and a plane - definition, examples of finding.

Lecture






  Angle between a straight line and a plane - definition, examples of finding.

The line a intersects the plane α. and not perpendicular to the plane. The bases of the perpendiculars dropped from the points of the straight line a to the plane α lie on the straight line a '. This straight line is called the projection of the straight line a on the plane α.
The angle between the straight line and the projection of this straight line onto a plane is called the angle between the straight line and the plane .

We begin this article with the definition of the angle between a straight line and a plane. After this, we will show how the angle between the straight line and the plane is found by the method of coordinates, analyze in detail the solutions of typical examples and problems.

The angle between the line and the plane is the definition.

Before talking about the definition of the angle between a straight line and a plane, we recommend refreshing 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, we need several auxiliary definitions. We give these definitions.

Definition

A line and a plane intersect if they have one single common point, which is called the point of intersection of the line and the plane .

  Angle between a straight line and a plane - definition, examples of finding.

In this case, the straight line that intersects the plane can be perpendicular to this plane.

Definition

A straight line is perpendicular to a plane if it is perpendicular to any straight line lying in that plane.

  Angle between a straight line and a plane - definition, examples of finding.

Definition

The projection of the point M on the plane   Angle between a straight line and a plane - definition, examples of finding. is called either the point M itself, if M lies in the plane   Angle between a straight line and a plane - definition, examples of finding. , or the intersection point of the plane   Angle between a straight line and a plane - definition, examples of finding. and straight, perpendicular to the plane   Angle between a straight line and a plane - definition, examples of finding. and passing through the point M , if the point M does not lie in the plane   Angle between a straight line and a plane - definition, examples of finding. .

  Angle between a straight line and a plane - definition, examples of finding.

Definition

Projection of a line a onto a plane   Angle between a straight line and a plane - definition, examples of finding. call the set of projections of all points of the line a on the plane   Angle between a straight line and a plane - definition, examples of finding. .

Obviously, the projection of a straight line perpendicular to the plane   Angle between a straight line and a plane - definition, examples of finding. on the plane   Angle between a straight line and a plane - definition, examples of finding. is their point of intersection. It is also quite obvious that the projection of the line a , which intersects the plane   Angle between a straight line and a plane - definition, examples of finding. and not perpendicular to this plane, to the plane   Angle between a straight line and a plane - definition, examples of finding. is a straight line lying in a plane   Angle between a straight line and a plane - definition, examples of finding. and passing through the intersection of the line a and the plane   Angle between a straight line and a plane - definition, examples of finding. .

  Angle between a straight line and a plane - definition, examples of finding.

Now we have enough information to give a definition of the angle between a straight line and a plane.

Definition

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

The definition of the angle between the straight line and the plane allows us to conclude that the angle between the straight line and the plane is the angle between two intersecting straight lines: the straight line itself and its projection onto the plane. Therefore, the angle between a straight line and a plane is an acute angle.

  Angle between a straight line and a plane - definition, examples of finding.

The angle between the perpendicular lines and the plane is considered equal   Angle between a straight line and a plane - definition, examples of finding. , and the angle between parallel lines and planes is either not determined at all, or is considered equal   Angle between a straight line and a plane - definition, examples of finding. .

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Finding the angle between a straight line and a plane.

The conditions of the tasks in which you have to find the angle between a straight line and a plane are quite diverse. Depending on the source data, you have to choose the appropriate solution method. Signs of equality or similarity of figures, the cosine theorem and the definition of sine, cosine and tangent of an angle often help to cope with the task of finding the angle between a straight line and a plane. You can also find the angle between a straight line and a plane using the coordinate method. Let us dwell on it in more detail.

Suppose that in a three-dimensional space a rectangular coordinate system Oxyz is introduced, and the straight line a is defined in it, which intersects the plane   Angle between a straight line and a plane - definition, examples of finding. at point M and not perpendicular to the plane   Angle between a straight line and a plane - definition, examples of finding. and you need to find the angle   Angle between a straight line and a plane - definition, examples of finding. between the line a and the plane   Angle between a straight line and a plane - definition, examples of finding. .

Let's start with the initial data, from which we will repel when determining the angle between a straight line and a plane by the method of coordinates.

Direct a in a given rectangular coordinate system Oxyz corresponds to some equations of a straight line in space and a directing vector of a straight line in space, and the plane   Angle between a straight line and a plane - definition, examples of finding. - an equation of a plane of some kind and a normal vector of a plane. Let be   Angle between a straight line and a plane - definition, examples of finding. - directing vector of a ,   Angle between a straight line and a plane - definition, examples of finding. - normal plane vector   Angle between a straight line and a plane - definition, examples of finding. . So, we will assume that we know the coordinates of the direct vector of the line a and the coordinates of the normal vector of the plane   Angle between a straight line and a plane - definition, examples of finding. (if the equations of the line a and the plane are known   Angle between a straight line and a plane - definition, examples of finding. then the coordinates of vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. determined by these equations).

It remains to obtain a formula that will allow us to calculate the angle between a straight line and a plane from the known coordinates of the direct vector of the straight line and the normal vector of the plane.

Set aside vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. from the point of intersection of the line a and the plane   Angle between a straight line and a plane - definition, examples of finding. . Depending on the coordinates of the vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. There are four possible locations of these vectors relative to the given straight line and plane. Draw them on the drawing.

  Angle between a straight line and a plane - definition, examples of finding.

Obviously, if the angle between the vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. (denote it   Angle between a straight line and a plane - definition, examples of finding. ) sharp, then it complements the desired angle   Angle between a straight line and a plane - definition, examples of finding. between a straight line and a plane to a right angle, that is,   Angle between a straight line and a plane - definition, examples of finding. . If   Angle between a straight line and a plane - definition, examples of finding. then   Angle between a straight line and a plane - definition, examples of finding. .

Since the cosines of equal angles are equal, the last equalities can be written as follows:
  Angle between a straight line and a plane - definition, examples of finding.

Reduction formulas lead us to equalities   Angle between a straight line and a plane - definition, examples of finding. which after transformation take the form
  Angle between a straight line and a plane - definition, examples of finding.

That is, 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 plane .

In the section on finding the angle between two vectors, we found that the angle between the vectors is equal to the ratio of the scalar product of vectors and the product of the lengths of these vectors, then the formula for calculating the sine of the angle between the straight line and the plane is   Angle between a straight line and a plane - definition, examples of finding. .

Therefore, the formula for calculating the angle between a straight line and a plane according to the coordinates of the direct vector of the straight line and the normal vector of the plane is   Angle between a straight line and a plane - definition, examples of finding. .

The basic trigonometric identity allows us to find the cosine of an angle with a known sine. Since the angle between the straight line and the plane is sharp, the cosine of this angle is a positive number and is calculated by the formula   Angle between a straight line and a plane - definition, examples of finding. .

Now we can find the sine of the angle, the cosine of the angle and the angle itself between the straight line and the plane using the formulas obtained. Let's solve some typical examples.

Example.

Find the angle, sine and cosine of the angle between the line   Angle between a straight line and a plane - definition, examples of finding. and the plane   Angle between a straight line and a plane - definition, examples of finding. .

Decision.

The canonical equations of a straight line in space allow us to immediately obtain the coordinates of the directing vector — they are given by numbers in the denominators of fractions. I.e,   Angle between a straight line and a plane - definition, examples of finding. - directing vector straight   Angle between a straight line and a plane - definition, examples of finding. .

The general equation of the plane contains the coordinates of the normal vector of the plane in the form of coefficients for variables x , y and z . That is, the normal plane vector   Angle between a straight line and a plane - definition, examples of finding. is a vector   Angle between a straight line and a plane - definition, examples of finding. .

Substitute the coordinates of vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. in the formula for calculating the sine of the angle between a straight line and a plane:
  Angle between a straight line and a plane - definition, examples of finding.

Then   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. .

Answer:

  Angle between a straight line and a plane - definition, examples of finding.

Example.

On vectors   Angle between a straight line and a plane - definition, examples of finding. built a pyramid. Find the angle between the straight line AD and the plane ABC .

Decision.

To calculate the angle between a straight line and a plane using the obtained formula, we need to know the coordinates of the direct vector of the straight line and the normal vector of the plane. The direction vector of the straight line AD is the vector   Angle between a straight line and a plane - definition, examples of finding. .

Normal vector   Angle between a straight line and a plane - definition, examples of finding. plane abc perpendicular and vector   Angle between a straight line and a plane - definition, examples of finding. and vector   Angle between a straight line and a plane - definition, examples of finding. that is, as a normal vector of the ABC plane, one can take the vector product of vectors   Angle between a straight line and a plane - definition, examples of finding. and   Angle between a straight line and a plane - definition, examples of finding. :
  Angle between a straight line and a plane - definition, examples of finding.

It remains to substitute the coordinates of the vectors in the formula and calculate the required angle between the straight line and the plane:
  Angle between a straight line and a plane - definition, examples of finding.

Answer:

  Angle between a straight line and a plane - definition, examples of finding.


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Stereometry

Terms: Stereometry