Finding parametric equations for the line through the point that is perpendicular to

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

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.

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.