12. Lines in space





Lines can be specified in a variety of ways. One way is described as follows. Select a point P_{0} on the line l, and a nonzero vector v parallel to the line. The line l is then the unique line passing through P_{0} and parallel to v. Now P lies on l if and only if is parallel to v. As this condition is equivalent to r  r_{0} = tv for some scalar t. The equation
is called a parametric vector equation of the line l. (It is not unique, as a different point P_{0} on the line could have been chosen, changing r_{0}, and v can be replaced by any other nonzero vector parallel to l.) Each value of the parameter t determines a unique point P, with position vector r = r_{0} + tv, on the line l. As t takes all possible values, P takes all possible positions on the line l.


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