2.5. Lines In 3-Space And In 2-Space

Consider eq(2.1) for the case of 3-space.

Writing P=( p,q,r ) , A=( a,b,c )  and X( t )=( x,y,z ) , we have

        x=p+ta        y=q+tb         z=r+tc                                   (2.2)

which are known as scalar parametric equations, or simply, parametric equations, for the line.

For 2-space, only the 1st two equations in (2.2) are required.  The parameter t can then be eliminated to give

        xp a = yq b           Þ    b( xp )a( yq )=0          (2.3)

which is called a Cartesian equation for the line.  Provided a0 , we can put it in the point-slope form,

        yq= b a ( xp )

where the point ( p,q )  and slope b a  are easily discerned.

Eq(2.3) can also be written in terms of dot products, namely,

        N( XP )=0

where N=( b,a )  is called a normal vector to the line since it is perpendicular to the direction vector A, i.e., NA=baab=0 .  [cf. Fig.2.2]

Theorem 2.6

Let L be a line in R2 given by   N( XP )=0  and let   d= | PN | N .

Then X d     XL .

Furthermore, X     =d  iff X is the projection of P along N, i.e.,

X= PN NN N

Proof

Since NX=NP     XL , we have, by the Cauchy-Schwarz inequality,

| NP |    =    | NX | | XX || NN |   =     X N

Þ            X      | PN | N =d

Now, the equality holds iff X=αN  for some αR .  This means

        αNN=PN    Þ           α= PN NN                 Q.E.D.

Corollary

Let Q be a point not on L, then

  XQ     d   = | ( PQ )N | N     XL

where the equality applies iff XQ  is the projection of PQ  and d is called the distance of Q to L.