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Consider the line in $R^2$ that is given by the equation $d_1x_1 + d_2x_2 = c$ for numbers $d_1, d_2$ and $c$ in $R$ where $d_1$ and $d_2$ are not both zero. Find parametric equations of the line.

So I know a parametric equation for a line must have a form like:

$x_1 = T[d_1] + a_1$ and $x_2 = T[d_2] + a_2$ where $d_i$ is the direction vector and $a$ the translation vector. So, for the given equation I solve for $x_2$ like this:

$x_2 = \frac{c-d_1x_1}{d_2} = \frac{-d_1x_1}{d2} + \frac{c}{d_2}$ and start to see why are not both zero. But then, as one of them can be indeed zero, then it doesn't make sense to say that:

$x_1 = \frac{c-d_2x_2}{d_1}$

So I get stuck. I'm really trying hard to self-study linear algebra but this seems to much for me. Any help?

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up vote 3 down vote accepted

This is just one way to do it: first get a points on the line by setting $x_1$ to zero:

$$d_1 \cdot 0 +d_2 x_2 = c$$ $$ x_2 = c/d_2$$

so $$(0,c/d_2)$$ is a point on the line. The point $(c/d_1,0)$ is also on the line, and can be obtained by setting $x_2 = 0$. The difference of these points (treated as vectors) is $(c/d_1,-c/d_2)$ which will be a vector parallel to the line.

The line can then be writen as $\vec{p_0} + t\vec{v}$ where $\vec{p_0}$ is a point on the line, $\vec{v}$ is a vector parallel to the line, and $t$ is the parameter. Therefore, one possible parametrization is:

$$(x,y) = (c/d_1,0) + t\cdot (c/d_1,-c/d_2).$$

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