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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 $p_1 := (x_1 , y_1,z_1)$ 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 of a line passing through the 2 points $p_1 = (x_1, y_1, z_1)$ and $p_2 = (x_2 , y_2 , z_2)$. 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

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Oh well, I didn't get an edit notification again and did the same edit. – Gigili May 27 '12 at 19:11
Care to state the equations? – user31373 May 27 '12 at 19:12
up vote 0 down vote accepted

The first form expresses a line in terms of a point it passes through and its direction. The second form expresses a line in terms of two points it passes through.

The forms are equivalent. To convert the second form to the first, simply calculate $p_2 - p_1$ to get the direction vector. You can use either $p_1$ or $p_2$ as the point that the line passes through.

To convert the first form to the second, take $p_2 = (x_1 + a, y_1 + b, z_1 + c)$.

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