Next: Dot product.
Up: Analytic geometry in the
Previous: Analytic geometry in the
The euclidean 3dimensional space will be denoted througout these pages by
.
A coordinate system is given in
;
the origin is denoted by O, the axes are perpendicular
and respective unit vecors on the axes are
,
and
.
Figure 1:
The point
M(2,3,3/2).

Every plane in the 3dimensional space has an equation of the form
ax+by+cz+d=0
This equation is called a cartesian equation of the plane.
For example, a cartesian equation of the plane through the points A(2,0,0), B(0,3,0) and C(0,0,2) is
3x+2y+3z6=0
Proposition 1.1
Let
be a plane with cartesian equation
ax+
by+
cz+
d=0.
 If a=0, the plane is parallel to the xaxis.
 If b=0, the plane is parallel to the yaxis.
 If c=0, the plane is parallel to the zaxis.
 If d=0, the plane lies on the origin.
Corollary 1.2
Let
be a plane with cartesian equation
ax+
by+
cz+
d=0.
 If a=b=0, the plane is parallel to the xyplane.
 If b=c=0, the plane is parallel to the yzplane.
 If a=c=0, the plane is parallel to the xzplane.
Figure 2:
A plane.

Take two planes
and
whose respective cartesian equations are:
Then:
 1.

and
are parallel if, and only if, the vectors
(a_{1},b_{1},c_{1}) and
(a_{2},b_{2},c_{2}) are proportional.
 2.

and
are identical if, and only if, the vectors
(a_{1},b_{1},c_{1},d_{1}) and
(a_{2},b_{2},c_{2},d_{2}) are proportional.
 3.

and
are perpendicular if, and only if, the vectors
(a_{1},b_{1},c_{1}) and
(a_{2},b_{2},c_{2}) are orthogonal (v.s. 1.12).
Example 1.3
 1.
 The planes whose respective cartesian equations are
2x+3yz+1=0 and
4x+6y2z5=0 are parallel and distinct.
 2.
 The planes whose respective cartesian equations are
2x+3yz+1=0 and
x2y4z+3=0 are perpendicular.
For more details on orthogonal vectors, please go to the tutorial in Linear Algebra.
Figure 3:
Respective position of two planes.

If two planes are not parallel, their intersection is a (straight) line. Conversely, a line can be given by not less
than two cartesian equations.
Example 1.4
Let
and
.
their intersection is defined by the system of two linear equations:
Solving this sytem, we get:
In fact we replaced two planes in general position by two planes, one of them parallel to the
yaxis, the second
one parallel to the
xaxis. With cartesian equations, we cannot do anything better.
The respective positions of a plane and a line are as follows:
 If their intersection is empty, they are parallel (see Fig. 4(a));
 If their intersection is a singleton, i.e. contains a single point, the line is a secant to the plane
(see Fig. 4(b));
 The line can be included into the plane; their intersection is the line itself
(see Fig. 4(c)).
Figure 4:
Respective position of a line and a plane.

Next: Dot product.
Up: Analytic geometry in the
Previous: Analytic geometry in the
Noah DanaPicard
20010530