Consideration in the geometry of lines and planes involves the study of the question of their mutual arrangement. In this article, we give a definition of a perpendicular and inclined to a plane and a straight line, and also consider the properties that they possess.
Equations for Plane and Line
It is impossible to proceed to the consideration of the question of a perpendicular and inclined to different geometric objects until the mathematical methods of specifying these objects are studied. We are talking about the types of equations to a plane and a straight line.
A straight line in space, two-dimensional and three-dimensional, is often given in the form of a vector equation. It has the form:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + λ * (a; b; c)
In the first brackets of the right side of the equality are the coordinates of the point that belongs to the line. In the second brackets are the coordinates of the vector, which is the guide for this geometric object. The parameter λ can take an arbitrary value.
It is easy to work with this equation in the three-dimensional case. In the two-dimensional case, it can also be used (only without the z coordinate). However, in the coordinate plane, the line is often set in a different form:
A * x + B * y + C = 0
Here A, B, C are some numbers, and (A; B) is a vector that is perpendicular to the line. This equation is called general.
A plane as a two-dimensional object in space is most often specified in a general form, that is:
A * x + B * y + C * z + D = 0
It is important not to confuse this equation with the previous one. Here the vector, which is given by the coordinates (A; B; C), turns out to be normal (perpendicular) to the plane, that is, it is its directing vector.
Perpendicular for straight and plane
The study of the oblique and perpendicular to the line and plane will begin with a perpendicular vector. It has already been said above that for a straight line on a plane and for a plane in three-dimensional space, which are given in the form of an equation of a general form, it is not difficult to determine the coordinates of a perpendicular vector. Its main property, which follows from the definition of the perpendicular, is that it with the considered geometric object makes an angle of 90 o .
The blue line in the figure below is perpendicular to the line shown.
Another property of the perpendicular to the line is that its scalar product with the direction vector will be zero.
The perpendicular to the plane, which is represented by a vector and a straight line, is shown in the figure below.
As in the case of a straight line, the scalar product of this perpendicular to any vector belonging to the plane will be zero.
If the perpendicular is not known for the plane, then it is easy to find. For this, it is necessary to calculate the vector product of any two non-parallel vectors belonging to the plane.
Finding a perpendicular to a plane and a straight line is important because it allows you to calculate the distance from these objects to a certain point in space.
Inclined to a straight line
We continue to consider the properties of the perpendicular and inclined to the line and plane. We first give the definition of a slant for a straight line. It means any line that intersects a given line at a certain angle, not equal to 90 o .
If you drop the perpendicular from another point of the inclined line onto another line, then the segment formed by the point of the perpendicular and the intersection point of the objects will be the projection of the segment inclined onto the line.
Calculating the angle φ between intersecting lines is quite simple. To do this, use the formula as follows:
φ = arccos (| (v¯ * u¯) | / (| v¯ | * | u¯ |))
Here v¯ and u¯ are the direction vectors of the lines under consideration.
For the case on the plane, this formula can always be used without preliminary calculations. In space, it should be borne in mind that the lines can form a certain angle between themselves, but at the same time not intersect (be crossed). In this case, you should first check whether the lines have a common intersection point.
Inclined to the plane
In the issue of perpendicular and inclined, it remains to consider inclined to the plane.
Any line that intersects a given plane at some acute angle is inclined to it. If each point of the line is projected onto the plane, drawing the corresponding perpendiculars to it, then we get the projection of the entire line. The angle between this projection and the original straight line is the intersection angle of the objects in question.
The figure below shows the line r, which is inclined to the plane “pi”.
Line r 'is the projection of r. The angle between pi and r is denoted by α. To determine it, use the following formula:
α = arcsin (| (v¯ * n¯) | / (| v¯ | * | n¯ |))
Where v¯ and n¯ are the direction vectors of the line r and the plane “pi”, respectively (see. Fig.).
Task example
We show how you can use the acquired knowledge about the perpendicular and inclined to solve the following problem:
The equations of the line and plane are given:
(x; y; z) = (1; 0; 2) + λ * (1; 1; 1);
x + 2 * y - 1 = 0
The given line is inclined to the plane. It is necessary to determine at what angle they intersect.
We write out the coordinates of the corresponding direction vectors:
for a straight line: v¯ = (1; 1; 1);
for the plane: n¯ = (1; 2; 0)
Substitute these coordinates in the formula for the angle α, we obtain:
α = arcsin (| ((1; 1; 1) * (1; 2; 0)) | / (| (1; 2; 0) | * | (1; 1; 1) |)) =
= arcsin (3 / (√5 * √3)) ≈ 50.77 o
Thus, the line, being inclined to the plane, intersects it at an angle of 50.77 o .