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Planes and lines.

The euclidean 3-dimensional space will be denoted througout these pages by $\mathcal{E}$. A coordinate system is given in $\mathcal{E}$; the origin is denoted by O, the axes are perpendicular and respective unit vecors on the axes are $\overrightarrow{i} $, $\overrightarrow{j} $ and $\overrightarrow{k} $.


  
Figure 1: The point M(2,3,3/2).
\begin{figure}
\mbox{\epsfig{file=axes.eps,height=4cm} }
\end{figure}

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 $\mathcal{P}$ be a plane with cartesian equation ax+by+cz+d=0.

Corollary 1.2   Let $\mathcal{P}$ be a plane with cartesian equation ax+by+cz+d=0.


  
Figure 2: A plane.
\begin{figure}
\mbox{\epsfig{file=Aplane.eps,height=4cm} }
\end{figure}

Take two planes $\mathcal{P_1}$ and $\mathcal{P_2}$ whose respective cartesian equations are:

\begin{displaymath}a_1x+b_1y+c_1z+d_1=0 \qquad \text{and} \qquad a_2x+b_2y+c_2z+d_2=0
\end{displaymath}

Then:
1.
$\mathcal{P_1}$ and $\mathcal{P_2}$ are parallel if, and only if, the vectors (a1,b1,c1) and (a2,b2,c2) are proportional.
2.
$\mathcal{P_1}$ and $\mathcal{P_2}$ are identical if, and only if, the vectors (a1,b1,c1,d1) and (a2,b2,c2,d2) are proportional.
3.
$\mathcal{P_1}$ and $\mathcal{P_2}$ 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.


   
Figure 3: Respective position of two planes.
\begin{figure}
\mbox{\subfigure[parallel]{\epsfig{file=ParallelPlanes.eps,height...
...gure[intersecting]{\epsfig{file=IntersectLine.eps,height=3cm} }
}\end{figure}

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 $\mathcal{P}_1: \; 2x-y+z=1$ and $\mathcal{P}_2: \; x-y+2z=2$. their intersection is defined by the system of two linear equations:

\begin{displaymath}\begin{cases}
2x-y+z=1 \\ x-y+2z=2
\end{cases}\end{displaymath}

Solving this sytem, we get:

\begin{displaymath}\begin{cases}
x-z =-1 \\ y-3z=-3
\end{cases}\end{displaymath}

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:


    
Figure 4: Respective position of a line and a plane.
\begin{figure}
\mbox{\subfigure[parallel]{\epsfig{file=LinePlane1.eps,height=2cm...
...subfigure[included]{\epsfig{file=LinePlane3.eps,height=1.5cm} }
}\end{figure}

Example 1.5  


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