Consider eq(2.1) for the case of 3-space.
Writing , and , we have
which are known as scalar parametric equations, or simply, parametric equations, for the line.
For 2-space, only the 1st two equations in (2.2) are required. The parameter t can then be eliminated to give
which is called a Cartesian equation for the line. Provided , we can put it in the point-slope form,
where the point and slope are easily discerned.
Eq(2.3) can also be written in terms of dot products, namely,
where is called a normal vector to the line since it is perpendicular to the direction vector A, i.e., . [cf. Fig.2.2]
Let L be a line in R2 given by and let .
Since , we have, by the Cauchy-Schwarz inequality,
Now, the equality holds iff for some . This means
Let Q be a point not on L, then
where the equality applies iff is the projection of and d is called the distance of Q to L.