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.


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 ) } .


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.


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.


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.