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.

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

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

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

The angle between the perpendicular lines and the plane is considered equal , and the angle between parallel lines and planes is either not determined at all, or is considered equal .

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 at point M and not perpendicular to the plane and you need to find the angle between the line a and the plane .

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 - an equation of a plane of some kind and a normal vector of a plane. Let be - directing vector of a , - normal plane vector . 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 (if the equations of the line a and the plane are known then the coordinates of vectors and 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 and from the point of intersection of the line a and the plane . Depending on the coordinates of the vectors and There are four possible locations of these vectors relative to the given straight line and plane. Draw them on the drawing.

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

Since the cosines of equal angles are equal, the last equalities can be written as follows:

Reduction formulas lead us to equalities which after transformation take the form

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 .

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 .

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 .

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.