Finding Parametric Equations for Line: Perpendicular vs Parallel

[coleko]

Finding parametric equations for the line through the point that is perpendicular to plane and parallel?

What is the difference when finding parametric equations for a line through a point that is perpendicular vs. parallel? Surely there must be some difference but I cannot seem to figure it out.

Here is an example

Find parametric equations for the line through the point (2,4,6) perpendicular to plane x-y+3x=7
Also find parametric equations for the same line parallel to the same plane...

Do you find them both by using r(t)=r+ t*v
?

 

[Simon Bridge]

Finding parametric equations for the line through the point that is perpendicular to plane and parallel?

What is the difference when finding parametric equations for a line through a point that is perpendicular vs. parallel? Surely there must be some difference but I cannot seem to figure it out.

They go in different directions.

Note, there is only one line through a point that is perpendicular to a surface, and an infinite number of possibilities parallel.

Here is an example

Find parametric equations for the line through the point (2,4,6) perpendicular to plane x-y+3x=7
Also find parametric equations for the same line parallel to the same plane...

Do you find them both by using r(t)=r+ t*v
?

Some issues with how you phrased that... what you have provided is not the equation of a plane (oh I suppose it could be - it would be 4x-y=7 for any z - so the plane does not intersect the z-axis) ... and "the same line" cannot be parallel. But yes you use the equation of a line both times - but that is not all you do. If r is the point then v must be a vector pointing in some direction: how do you determine that direction?

 

[coleko]

To find the direction of the vector you would take the coefficients of the plane which would be <1,-1,3> I apologize because the 3x I wrote should really be 3z and the equation should read x-y+3z=7

I also realize that the same line cannot be parallel and perpendicular at the same time I am just questioning my methods on how to figure those out...

 

[Simon Bridge]

Well, a line through point P at position $\vec{p}$ perpendicular to a plane $ax+by+cz+d=0$ has equation: $\vec{r}(t)=\vec{p}+\vec{n}t$ where $\vec{n}=(a,b,c)$ is the normal to the plane.

A line parallel to the same plane would be and of $\vec{r}(t)=\vec{p}+(a\vec{v}+b\vec{n}\times\vec{v})t$ where $\vec{v}$ is any vector in the plane and $a$ and $b$ are arbitrary scalars.

See the difference?

 

[HallsofIvy]

Note that, in three dimensions, there exist a single line through a given point, perpendicular to a given plane but there exist and infinite number of lines through a given point parallel to the given plane.