Page 66 - 4263

Analytic geometry in space

−→
(p) (fig. 8.2). Vector n is called the normal vector of plane (p). In order to obtain the equation
−−−→
of the plane, lets select any point M(x, y, z) on it and get vector M 0 M = (x−x 0 , y−y 0 , z−z 0 ).
−−−→ − → −→
−→
Regardless of the position of point M vectors n and M 0 M are perpendicular: n · M 0 M = 0.
Let’s write down the scalar multiplication in the view of coordinates:
A · (x − x 0 ) + B · (y − y 0 ) + C · (z − z 0 ) = 0. (8.3)

This is the equation of the plane, that passes through the given point M 0 (x 0 , y 0 , z 0 ) and has
− →
the normal vector n = (A; B; C). Therefore, a plane is matching with the equation of the first
order relatively to variables x, y and z.

8.2.2. General equation of a plane
Having opened the brackets in 8.3 and having marked D = Ax 0 − By 0 − Cz 0 as the free term,
we will get the equation:
Ax + By + Cz + D = 0 (8.4)
which is called the general equation of the plane.
The equation of the first order matches any plane. Now let’s prove that any equation Ax +
By +Cz +D = 0 of the first order relatively to the coordinates x, y and z is the equation of the
plane. Suppose at least one of coefficients A, B, C isn’t equal to zero. Let it be coefficient C.
D
Then the general equation can be written in such a way: A·(x−0)+B·(y−0)+C·(z+ ) = 0.
C
Having compared it with equation (8.3), the following conclusion can be made: this is the plane
−→
D
with normal vector n = (A; B; C) passing through the given point M 0 (0, 0, − ).
C
Lets study general equation 8.4, i.e. look through the cases when it may not be complete.
1. If D = 0 than the equation looks like: Ax + By + Cz = 0 It is obvious that
this plane passes through the beginning of coordinates, as point O(0, 0, 0)
satisfies this equation.
2. Assume the first coefficient A = 0. Then the equation By + Cz + D = 0 will
describe the plane, passing parallel to axis Ox. Indeed, its normal vector
−→
n = (0; B; C) is perpendicular to axis Ox , so the plane is parallel to this
axis. Similar conclusions can be made in case B = 0 or C = 0.
3. Assume A = 0 and D = 0. Than the equation By + Cz = 0 will describe the
plane which passes through the beginning of coordinates, and is parallel
to axis Ox, i.e. through the axis Ox itself. In case B = 0 and D = 0 the
plane Ax + Cz = 0 passes through axis Oy. In case C = 0 and D = 0 the
plane Ax + By = 0 passes through axis Oz.
4. Assume, that two out of three coefficients near x, y, z are equal to zero.
For example, A = 0 and B = 0. In this case the equation Cz + D = 0 will
describe the plane which is parallel to axes Ox and Oy, that is parallel
− →
to plane OXY. Indeed, the normal vector n = (0; 0; C) in this case is
perpendicular to the plane, and at the same time is perpendicular to
plane (p), that is plane (p) must be parallel to the coordinate plane Oxy.
5. Assume that two out of three coefficients near x, y, z and a free term are
equal to zero. For example, A = 0, B = 0 and D = 0. Then the equation
Cx = 0 or x = 0 will describe plane Oxy. Indeed, the normal vector in this
− →
case n = (0; 0; 1)— is the normal vector to plane Oxy.
8.2.3. Equation of a plane passing through three given points

Let’s suppose we have got three points M 1 (x 1 , y 1 , z 1 ), M 2 (x 2 , y 2 , z 2 ), M 3 (x 3 , y 3 , z 3 ) on the plane
(p), which aren’t located on the same line (fig. 8.3).

66