2.2. Lines In n-Space

Rn is an analytic model of n-D Euclidean space En.  Thus, the geometric concepts and properties of En are expressed in terms of n-tuples of real numbers in Rn.

Definition (Point)

A point is a vector (n-tuple) in Rn.

Definition (Line)

Let P be a point and A a non-zero vector.

A line through P and parallel to A is the set of points

        L( P;A )={ P+tA:tR }   ={ P+tA }

The vector A is called the direction vector of the line.

A point Q is on the line L( P;A )  if Q=P+tA  for some t.

Properties

The line L( O;A )  is the linear span L( A )  of A.

Thus, we can write L( P;A )={ P+X:XL( A ) } .  [See Fig.2.1]