Many geometric shapes are formed by straight lines intersecting at right angles. For example, it is a square, a rectangle, a right triangle, or a straight quadrangular prism. In this article, we consider the question of the perpendicularity of two lines and the conditions that must be satisfied so that the line is perpendicular to the plane.
What equations are important to know?
The perpendicularity conditions of two straight lines and a straight line and a plane are not difficult to obtain if the corresponding equations for the named geometric objects are known.
The equation of any line both on the plane and in space can be written in universal vector form. For a three-dimensional case, it looks as follows:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + λ * (a; b; c)
Here, the variables x, z and y are the coordinates in the selected system, λ is any real number, and the triple of numbers (a; b; c) define a vector in the space called a guide (a straight line passing through a point with coordinates (x 0 ; y 0 ; z 0 )). This equation can be transformed into a general form, in canonical and parametric.
The plane is most conveniently represented in general form, which corresponds to the equation:
A * x + B * y + C * z + D = 0
Latin capital letters are coefficients. This expression can also be represented in vector, parametric forms and in the form of an equation in segments. The convenience of the given form of writing is that the first three coefficients correspond to the coordinates of a vector that is perpendicular to this plane, that is:
n¯ (A; B; C) is the direction vector of the plane
Perpendicularity of two straight lines
The condition of perpendicularity of the lines is not difficult to understand, for this it is enough to establish whether their directing vectors are perpendicular. The latter can be clarified by calculating the scalar product. Suppose that v¯ and u¯ are direction vectors for two lines. If the latter are perpendicular, then:
(v¯ * u¯) = 0
This condition of perpendicularity of two lines is a must. Nevertheless, it will be sufficient only for the case of two-dimensional space. In three-dimensional space, in addition to this expression, the distance between the lines should also be calculated. If the equality above holds, and the specified distance is zero, then the lines will intersect at an angle of 90 o , that is, they will be perpendicular.
To calculate the distance d between lines in space, use the expression:
d = | [M 1 M 2 ¯ * u¯] | / | u¯ |
Here M 1 M 2 ¯ is a vector built on two points, each of which belongs to the corresponding line (M 1 lies on the first line, M 2 on the second).
Plane and straight
The perpendicularity condition for these objects is as follows:
u¯ = k * n¯
In other words, the line will intersect the plane at an angle of 90 o only when its guide vector is parallel to the normal to the plane. The fact of parallelism means that the vector of the straight line u¯ can be obtained by multiplying the vector n¯ normal to the plane by some specific number k.
There are also other ways to find out if the vectors u¯ and n¯ are parallel. For example, if they are parallel, the angle between them must be zero, that is, the cosine of the angle calculated through the scalar product will be 1. In turn, the vector product of parallel vectors is zero.
Note that if the plane and the straight line are not given in the general and vector form, respectively, then we should bring them to these forms, and then use the given formulas for the perpendicularity conditions.