2.3. Some Simple Properties Of Straight Lines In Rn

Definition (Parallelism of Vectors)

Two vectors A and B are parallel iff A=αB  for some αR .

Theorem 2.1

Two lines L( P;A )  and L( P;B )  are equal iff A and B are parallel.

Proof of Þ

Let L( P;A )=L( P;B ) .  A point X on the line can therefore be written as

        X=P+αA=P+βB              for some α,βR

Hence,  A=γB  where γ= β α R .  That is, A and B are parallel.

Proof of Ü

Let A and B be parallel so that A=αB  for some αR .

If Q is on L( P;A ) , then Q=P+tA  for some tR .  Thus,

        Q=P+tA=P+tαB=P+βB       where β=tαR

Hence, Q is also on L( P;B ) .  Since this is true for arbitrary Q, we have L( P;A )L( P;B ) .  Reversing the role of A and B gives L( P;B )L( P;A ) .  Hence, L( P;A )=L( P;B ) .

Definition (Parallelism of Lines)

Two lines L( P;A )  and L( P;B )  are parallel if A and B are parallel.

Theorem 2.3 (Euclid's Parallel Postulate)

Given a line L and a point Q not on L, there is one and only one line L' through Q and parallel to L.

Proof

Let A be the direction vector of L.  We can write L'=L( Q,A ) .  According to theorem 2.1, L' is unique. Q.E.D.

Theorem 2.4 (2 distinct points determines a lines)

If PQ , there is 1 and only 1 line containing both P and Q.

The line can be written as { P+t( QP ) } .

Proof

The line L( P;QP )={ P+t( QP ) }  goes through both P and Q since they corresponds to t=0  and t=1 , respectively.

Let L' be another line that goes through both P and Q.  Since L' goes through P, we have L'=L( P;A )  for some nonzero A.  Now, L' also goes through Q so that P+αA=Q  for some α.  Hence, αA=QP  so that A and Q-P are parallel.  By theorem 2.2, L'=L( P;QP ) .  Q.E.D.

Example

From theoem 2.4, we see that Q is on L( P,A )  iff A and Q-P are parallel, i.e., QP=αA .

Consider the 3-space case with P=( 1,2,3 )  and A=( 2,1,5 ) .

To test if Q=( 1,1,4 )  is on L( P,A ) , we examine QP=( 0,1,1 )  and find that it is not a multiple of A.  Hence, Q is not on the line.

For Q=( 5,0,13 ) , we have QP=( 4,2,10 )=2A  so that Q is on L.

Theorem 2.5 (Linear Dependence)

Two vectors A,B R n  are linearly dependent iff they lie on the same line through the origin.

Proof

If either vector is zero, the theorem holds trivially.

If both are nonzero, they are dependent iff B=αA  for some αR .

However, B=αA  iff B is on the line L( 0,A ) .  Q.E.D.