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Finding parametric equations for the line through the point that is perpendicular to 
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#1
Nov1512, 07:42 PM

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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 xy+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 ? 


#2
Nov1512, 08:11 PM

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


#3
Nov1512, 08:18 PM

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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 xy+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... 


#4
Nov1512, 09:01 PM

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


#5
Nov1612, 08:03 AM

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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.



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