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Equations of the plane

z
S

. 0 y

. . . . . . . . x

Figure 8.1 – Surface in the space

coordinates of which satisfy the equation (8.1).
A line in space can be considered as the line of crossing of two surfaces, i.e. the set of points,
that are common for both surfaces.
According to that, the line is determined by two given equations:
{
F 1 (x, y, z) = 0,
(8.2)
F 2 (x, y, z) = 0.
Equation (8.2) is called the equation of line (8.1) if coordinates of any point which belongs to
this line satisfy these equations and don’t satisfy if the point doesn’t belong to line (8.1).

8.2. Equations of the plane

Let’s show that a plane is the surface described by the first order equation relatively to coordi-
nates x, y and z. Depending on the way of representing a plane, equations can have different
view. Let’s consider these equations.

8.2.1. Equation of a plane, which passes through a fixed point and is perpendicular to the
given vector

−→
n
z
M

(p)
M 0

. . . . . . . . .
0 y

x

Figure 8.2 – Plane, which passes through a fixed point and is perpendicular to the given vector

− →
Let’s suppose that the position of plane (p) in space is determined by the given vector n =
(A; B; C), which is perpendicular to the plane, and point M 0 (x 0 , y 0 , z 0 ), which belongs to plane

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