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Equations of the plane

z
M
M 2 M 3

(p)
M 1

. . . . . . . . . .
0 y

x

Figure 8.3 – Plane passing through three given points

In order to create the equation of the plane, let’s take any point M on it and obtain vectors:
−−−→ −−−−→ −−−−→
M 1 M = (x − x 1 ; y − y 1 ; z − z 1 ), M 1 M 2 = (x 2 − x 1 ; y 2 − y 1 ; z 2 − z 1 ), M 1 M 3 = (x 3 − x 1 ; y 3 −
y 1 ; z 3 − z 1 ), which are complanar.
−−−→ −−−−→
According to the complanarity condition, their mixed product equals to zero: M 1 M ·M 1 M 2 ·
−−−−→
M 1 M 3 = 0 or in the view of coordinates:

x − x 1 y − y 1 z − z 1

x 2 − x 1 y 2 − y 1 z 2 − z 1 = 0. (8.5)

x 3 − x 1 y 3 − y 1 z 3 − z 1
This is the equation of the plane passing through three given points.
Remark 8.1. Having expanded the determinant by elements of the first row we
will reduce this equation to the general view.

8.2.4. Equation of a plane in segments

z

M 3
c

(p)
M y
. . . . . . . . . . . . . b
0
a M 2

x M 1

Figure 8.4 – Plane in segments

Let’s suppose we have segments a, b and c, cut by the plane (p) on the coordinate axes
(fig. 8.4). These segments determine the only plane. Thus, we have three points M 1 (a, 0, 0),
M 2 (0, b, 0), M 3 (0, 0, c) on coordinate axes that belong to the same plane. Let’s use the equation

x − a y z

(8.5), having substituted these points’ coordinates into it: −a b 0 = 0 or (x − a)bc +

−a 0 c
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