2.8. Planes And Vector-Valued Functions

A plane M={ P+sA+tB }  may be considered as a mapping

        X: R 2 M

                ( s,t )    X( s,t )=P+sA+tB                   (2.6)

where X is called a vectored valued function of 2 real variables.  The scalars s and t are called parameters and eq(2.6) is called a parametric, or vector, equation of the plane.  For vectors in Rn, eq(2.6) represents n scalar equations.  Taking as example the case R3 and write

        P=( p 1 , p 2 , p 3 )         A=( a 1 , a 2 , a 3 ) B=( b 1 , b 2 , b 3 )           X=( x,y,z )

we have

        x= p 1 +s a 1 +t b 1               y= p 2 +s a 2 +t b 2              z= p 3 +s a 3 +t b 3

Eliminating s and t gives a linear equation of the form

        ax+by+cz=d

known as the Cartesian equation of the plane.

Example

Let M={ P+sA+tB }  with P=( 1,2,3 ) , A=( 1,2,1 )  and B=( 1,4,1 ) .

The vector equation is

        X( s,t )=( 1,2,3 )+s( 1,2,1 )+t( 1,4,1 )

which is equivalent to 3 scalar parametric equations

        x=1+s+t               y=2+2s4t          z=3+st

To obtain the Cartesian equation, we solve s, t from the 1st and 3rd eqs

        s+t=x1

        st=z3

to get

        s= 1 2 ( x+z4 )                t= 1 2 ( xz+2 )

Substituting them into the 2nd eq gives

        y=2+( x+z4 )2( xz+2 )

or             y=6x+3z