# Thread: Equation of a line in space

1. ## Equation of a line in space

If i have a line:
(-2-t, 1+2t, 3t)

How do I get a,b,c?

so.. how can I rewrite in the form ax + by = c

2. Originally Posted by Nforce
If i have a line:
(-2-t, 1+2t, 3t)

How do I get a,b,c?

so.. how can I rewrite in the form ax + by = c
You can't. In three dimensions, for any a, b, or c, the equation ax+ by= c represents a plane not a line! Notice, in fact, that your equation, ax+ by= c, says nothing about "z". In three dimensions, ax+ by= c is a plane parallel to the z-axis.

One form you could use is this: solve x= -2- t, y= 1+ 2t, and z= 3t for t: t= -2- x, t= (y-1)/2, and t= z/3. Now set those all equal:
$\displaystyle -2-x= \frac{y- 1}{2}= \frac{z}{3}$. That is called the "symmetric form" for the line.

But again- a line in 3 dimensions cannot be written as a single equation. Essentially, each equation allow you to solve for one variable in terms of the others and so reduces the dimension by 1. In 2 dimensions, one equations reduces to 2- 1= 1 dimension- a line if the equation is linear or, more generally, a curve. But in 3 dimensions, one equation reduces to 3- 1= 2, a plane if the equations are linear or, more generally, a surface. If you have two equations, as in $\displaystyle -2-x= \frac{y-1}{2}$ and $\displaystyle \frac{y- 1}{2}= \frac{z}{2}$ then you reduce to 3- 2= 1 dimension, a line if the equations are linear or, more generally, a curve.

3. But it is true that $\displaystyle \dfrac{x+2}{-1}=\dfrac{y-1}{2}=\dfrac{z}{3}$ is the symmetric form of the line.