Thread: parametric description of plane help

if c is the plane through the following points
(0,1,1) (0,1,0) and (-2,-1,-1)

how can I find an equation and parametric description of the plane?

I know the equation of the plane can be written as

a(x-x0) + b(y-y0) +c(z-z0) for a point (x0,y0,z0) and a normal vector (a,b,c)

how can I find the normal vector from the points?
I am also not sure about the difference the equation and parametric representation

thanks.

The vector from point to is . So two vectors in the plane are the vector from (0, 1, 1) to (0, 1, 0), <0- 0, 1- 1, 0- 1>= <0, 0, -1> and the vector from (0, 1, 1) to (-2, -1, -1), <0-(-2), 1-(-1), 1-(-1)>= <2, 2, 2>.

A vector perpendicular to those two vectors, and so perpendicular to the plane is their cross product.

A two dimensional surface in a three dimensional space can be written as a single equation, z= f(x,y) or, more generally, g(x,y,z)= constant. A plane can be written as a linear equation, say, ax+ by+ cz= d for some constants, a, b, c, and d. Parametric equations, in three dimensions, are three equations x= f(t), y= g(t), z= h(t) for a one dimensional figure (a curve or line) or x= f(s, t), y= g(s,t), z= h(s,t) for a two dimensional figure (the number of parameters is the dimension). A plane can be written in linear equations: x= as+ bt+ c, y= ds+ et+ f, z= gs+ ht+ i.

Given a surface given by the single equation, z= f(x,y), you can always take x and y themselves as parameters: x= s, y= t, z= f(s, t).

thanks for the help. I managed to get the normal vector as (2,-2,0)

I have another question


I have a normal vector (2,-2,0) of the plane and a point on a line orthogonal to the plane. (4,0,1). So my goal is to try and find the parametric description of this line.
now to do that I need a vector v, but how can I find this vector with the given info?

the parametric representation is given by p + tv where p is a point and v is a vector. I need another point on this line to find the vector. Since this line is orthogonal to the plane I know that this vector shoukd be parallel to the normal vector of the plane. But im having difficulty trying to find another point on this line.