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Home Vectors Vectors and Lines in Space Theorem 2
 
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Theorem 2

The next theorem shows that a line in space is uniquely determined by a position vector and a direction vector. That is, if two lines L and M have a position vector and direction vector in common, then L and M must be the same line.

THEOREM 2

Given a vector P and a nonzero vector D, the line X = P + tD is the unique line with position vector P and direction vector D.

PROOF

Let L be the line X = P + tD. Setting t = 0 and t = 1 we see that P and P + D are position vectors of L, so D is a direction vector of L.

Let X = Q + sE be any line M with position vector P and direction vector D. We show X = Q + sE is another vector equation for L. For some s0,

P = Q + s0E.

Also, D = B - A for some position vectors of M,

A = Q + s1E, B = Q + s2E.

Thus 

D = (Q + s2E) - (Q + s1E) = (s2 - s1)E.

Since D ≠ 0, s2 - s1 ≠ 0. Thus the following are equivalent:

X = P + tD

for some t,

X = Q + s0E + t(s2 - s1)E

for some t,

X = Q + (s0 + ts2 - ts1)E

for some t,

X = Q + sE

for some s.

Home Vectors Vectors and Lines in Space Theorem 2

Last Update: 2006-11-05