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# Planes and lines.

The euclidean 3-dimensional 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 .

Every plane in the 3-dimensional 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+3z-6=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 x-axis.
• If b=0, the plane is parallel to the y-axis.
• If c=0, the plane is parallel to the z-axis.
• 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 xy-plane.
• If b=c=0, the plane is parallel to the yz-plane.
• If a=c=0, the plane is parallel to the xz-plane.

Take two planes and whose respective cartesian equations are:

Then:
1.
and are parallel if, and only if, the vectors (a1,b1,c1) and (a2,b2,c2) are proportional.
2.
and are identical if, and only if, the vectors (a1,b1,c1,d1) and (a2,b2,c2,d2) are proportional.
3.
and are perpendicular if, and only if, the vectors (a1,b1,c1) and (a2,b2,c2) are orthogonal (v.s.  1.12).

Example 1.3

1.
The planes whose respective cartesian equations are 2x+3y-z+1=0 and 4x+6y-2z-5=0 are parallel and distinct.
2.
The planes whose respective cartesian equations are 2x+3y-z+1=0 and x-2y-4z+3=0 are perpendicular.

For more details on orthogonal vectors, please go to the tutorial in Linear Algebra.

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 y-axis, the second one parallel to the x-axis. 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)).

Example 1.5

Next: Dot product. Up: Analytic geometry in the Previous: Analytic geometry in the
Noah Dana-Picard
2001-05-30