School of Mathematics and Statistics, The University of Sydney
 12. Lines in space
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The parametric vector equation

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Lines can be specified in a variety of ways. One way is described as follows.

Select a point P0 on the line l, and a non-zero vector v parallel to the line.

The line l is then the unique line passing through P0 and parallel to v.

         z                              P                     r-  r0              P0            r                r0       l                     v          O                         y     x

Now P lies on l if and only if ----> P0P is parallel to v. As

--P-->P =  -O-->P  - -O-P-->  = r - r ,  0              0        0

this condition is equivalent to r - r0 = tv for some scalar t. The equation

r- r0 = tv

is called a parametric vector equation of the line l. (It is not unique, as a different point P0 on the line could have been chosen, changing r0, and v can be replaced by any other non-zero vector parallel to l.)

Each value of the parameter t determines a unique point P, with position vector r = r0 + tv, on the line l. As t takes all possible values, P takes all possible positions on the line l.

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