I have been searching the internet and can not find how to do this. If given a point and a plane, how would I find the equation for a line that passes through the point and is parallel to the plane?


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One way to find the equation of such a line is to use the normal vector of your plane and write the equation of a plane parallel to your plane and passing through the given point.

Any line on this new plane is parallel to the original plane.

Pick any new point on the new plane and write the equation of the line passing through this new point and the original given point.

This line has both required properties.


Denote the line and plane by $\ell$ and $P$, respectively.

To get an equation of $\ell$, you need:

  • a point on $\ell$
  • a direction vector for $\ell$ (i.e., a vector parallel to $\ell$).

You already know a point on $\ell$. Now, use an equation of $P$ to get two points $A$ and $B$ on $P$. Then $\overrightarrow{AB}$ is parallel to $P$, hence is parallel to $\ell$. So $\overrightarrow{AB}$ is a direction vector for $\ell$.

Finally, use the given point on $\ell$ and the direction vector $\overrightarrow{AB}$ for $\ell$ to get an equation for $\ell$ (or a set of parametric equations for $\ell$).


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