In our book of analytic geometry we have a title The canonical form of a line it is the equation of a line passing through a point p1(x1 , y1,z1) and parallel to a vector whose Direction Ratio is a:b:c under another title The Symmetric (Two point) form of a line is the equation pf line passing through the 2 points p1 (x1, y1, z1) & p2 (c2 , y2 , z2) so what is the difference between both of them ?? I can figure out the difference between them and the parametric form but those two can't get it ???!! am so confused

I believe a line can be fixed by saying either: 1. it passes through some point and is parallel to another line, or, 2. it passes through two points in space

so, if you are given just one point in space, then there would be an infinite number of lines tat you could construct passing through that point

in order to /fix/ it to one particular line, you either need to say is has to be parallel to some other vector (or line), or that it also has to pass through another fixed point

do you follow?

I mean that either way we get the same equation why are the having two names??

there are two names because they describe two different methods of getting to the equation of a line. think of quadratic equations as an example. you can solve these by either factoring or by using the equation for the solution to a general quadratic equation. in both cases you end up with the same solutions.

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