# Find the equation of a plane perpendicular to a line and goes through a point

## Homework Statement

find equation of plane P that is perpendicular to line L which passes through the point (-2,-2,3)

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## The Attempt at a Solution

[/B]line L passes through the points (1,2,1) and (0,0,-3) I have worked out the parametric equations of line L to be
x=1-t
y=2-2t
z=1-4t
i know that if dot product of two vectors = 0 then they are perpendicular if that helps. im stuck on how to piece this question together.
any help would be greatly appreciated

haruspex
Homework Helper
Gold Member
2020 Award
if dot product of two vectors = 0 then they are perpendicular
Right, so you need a vector in the direction of the line.

cathal84
Right, so you need a vector in the direction of the line.
i found a directional vector on the line it is <-1,-2,-4> so if i do dot product of that vector with <1,-1/2,0> (i just came up with in my head thinking what vector do i need to have that will give me dot product equal to 0) this will give me the dot product = 0.
so can i conclude that the vector <1,-1/2,0> is on the plane?

i found a directional vector on the line it is <-1,-2,-4> so if i do dot product of that vector with <1,-1/2,0> (i just came up with in my head thinking what vector do i need to have that will give me dot product equal to 0) this will give me the dot product = 0.
so can i conclude that the vector <1,-1/2,0> is on the plane?
even if so not sure how this is going to help me, exam
i found a directional vector on the line it is <-1,-2,-4> so if i do dot product of that vector with <1,-1/2,0> (i just came up with in my head thinking what vector do i need to have that will give me dot product equal to 0) this will give me the dot product = 0.
so can i conclude that the vector <1,-1/2,0> is on the plane?
even if that is true i dont know how that helps me

ehild
Homework Helper

if ##\vec r## is a position vector of the plane than the difference ##\vec r - \vec r_0 ## is a vector lying in the plane. It is perpendicular to L. Write it out in components.

haruspex
Homework Helper
Gold Member
2020 Award
i found a directional vector on the line it is <-1,-2,-4> so if i do dot product of that vector with <1,-1/2,0> (i just came up with in my head thinking what vector do i need to have that will give me dot product equal to 0) this will give me the dot product = 0.
so can i conclude that the vector <1,-1/2,0> is on the plane?
You have to be careful to distinguish between a vector representing a direction of interest and a vector representing a point. If the plane does not pass through the origin then the vector for a point in the plane is not parallel to the plane. Vectors perpendicular to the given line will be parallel to the plane. See ehild's diagram.

member 545369
Alternatively:

You know that the line L is perpendicular to the plane. Also, you found that <-1,-2,-4> is a vector in the same direction as the line. Wouldn't that imply that <-1,-2,-4> is normal to the plane? Knowing the general formula of a plane ##ax+by+cz=d## , what do the coefficients ##a, b, ##& ##c## have to do with the normal to the plane?