2.4. Lines And Vector-Valued Functions In n-Space

The line L( P;A )  can be considered as the track of a moving particle with position X at time t given by

        X( t )=P+tA                   (2.1)

Here X( t )  is an example of a vector-valued function of a real variable, i.e.,

        X:RL( P,A ) R n     and tX( t )

The scalar t is often called a parameter and (2.1) a vector parametric equation, or simply, a vector equation of the line L( P;A ) .

Consider a line passing through points P and Q.  By theorem 2.4, it can be written as L=L( P;QP )  so that (2.1) becomes

        X( t )=P+t( QP )   =( 1t )P+tQ

Note that X( a )=X( b )

Û  ( 1a )P+aQ=( 1b )P+bQ

i.e.,          ( ab )( QP )=0

which means a=b  since QP=A0 .  Hence, 2 points on the line are distinct iff their parameters are distinct.

Consider 3 points X( a ) , X( b )  and X( c ) .  We say X( c )  is between X( a )  and X( b )  if a<c<b  or b<c<a .

A pair of points P and Q are congruent to another pair P' and Q' if PQ = P'Q' .  The norm PQ  is called the distance between P and Q.