# Thread: Finding a parametric equation to a line

1. ## Finding a parametric equation to a line

How would you find a parametric equation for a line passing through a point and perpendicular to a line passing through two other points?

Here is an example given to me, "Find parametric equations of the line passing through (3, -1, 3) and perpendicular to the line passing through (3, -2, 4) and (0, 3, 5)."

I have been reading around and I've only found information about how to find an equation to a plane but not a line. How would you approach this question? Any help would be appreciated! Thanks!

2. Originally Posted by capitol
How would you find a parametric equation for a line passing through a point and perpendicular to a line passing through two other points?

Here is an example given to me, "Find parametric equations of the line passing through (3, -1, 3) and perpendicular to the line passing through (3, -2, 4) and (0, 3, 5)."

I have been reading around and I've only found information about how to find an equation to a plane but not a line. How would you approach this question? Any help would be appreciated! Thanks!
First find the parametric equation of the line passing through (3,-2,4) and (0,3,5).

The equation of the line would be $\displaystyle x=3-3t$, $\displaystyle y=-2+5t$ $\displaystyle z=4+t$ (I leave it for you to verify this).

Now, $\displaystyle \mathbf{n}=\left<-3,5,1\right>$. The line perpendicular to this will have a vector that is perpendicular to $\displaystyle \mathbf{n}$. Thus, you need to find an $\displaystyle \mathbf{m}$ such that $\displaystyle \mathbf{m}\cdot\mathbf{n}=0$. Note that $\displaystyle \mathbf{m}$ is not unique!

So we can say, for example, $\displaystyle \mathbf{m}=\left<5,3,0\right>$.

Thus, the equation of our new line will be $\displaystyle x=3+5t$, $\displaystyle y=-1+3t$, $\displaystyle z=3$

Does this make sense?