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I need to find the equation of the line, so I want to use y-y1=m(x-x1) but for a vertical line the slope is undefined. So I am unsure how to proceed.

6 Answers 6

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How about below parameterization.

$ \rho = x \cos{\theta} + y \sin{\theta}$

where

$ \rho $ = perpendicular distance to the line from the origin (0,0)

$ \theta $ = angle of a line perpendicular to our line with the x axis

you can take a look at http://en.wikipedia.org/wiki/Hough_transform.

when line slope is $90^o$, $\theta = 0^o$, then $ \rho = x $.

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    when line slope is $0^o$, $\theta = 90^o$ and $\rho = y$.2011-04-14
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A vertical line in the plane is defined by the equation $x = c$, where $c$ is some real. This makes sense because along the vertical line, the $x$ value doesn't change.

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Here's a hint: there is one thing in common among the coordinates of all of the points on a vertical line. Express this common feature as an equation an you'll have the equation of your line.

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    You must read and interpret mathematics very carefully to succeed, especially once you hit algebra. "Where it hits x" should be "where it hits the x-axis", since the x-axis is the visual object that your line can hit. This place where it hits would be a point. If you read carefully what I wrote, I said that the common feature is among the COORDINATES of the points on the line. It might help to draw axes and label them with some coordinates, draw a vertical line, pick a few points on it and write down their coordinates. You will notice a pattern in their x-coordinates when you do this.2011-04-14
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You can also think of it using the slightly massaged one:

dx*(y-b) = dy*(x-a)

for the line passing through (a,b) and "having slope" dy/dx (here dx = 0 )

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use the equation of a line: $ax+by+c=0$

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Seeing as this is tagged as linear algebra:

You might be interested in expressing your line using a parametric equation involving vectors.

For example, if you want a vertical line (in the direction of the standard basis vector $(0,1)$) which passes through the point $(3,7)$, then you could use $(x,y)=(3,7)+t\cdot (0,1)$ where $t$ varies over all scalars (elements of $\mathbb{R}$ probably).

In this particular case of a vertical line, as others have noted, the equation would be equivalent to just writing $x=3+t\cdot 0=3$ since the $x$ value does not vary, and the $y$ value is not constrained.

More generally, the equation for the line in the direction of vector $(\Delta x,\Delta y)$ and passing through the point $(x_0,y_0)$ would be $(x,y) = (x_0,y_0) + t\cdot (\Delta x,\Delta y)$

You should understand this as an equation which describes the collection of points which includes $(x_0, y_0)$, along with any point we get by starting at $(x_0, y_0)$ and moving some distance (forward or backward) in the direction of $(\Delta x, \Delta y)$. If you think about this a bit, you should see that this has the advantage of generalizing fairly easily to higher dimensional spaces, where you've got more variables than just $x$ and $y$.