Question

How do you find parametric equations for the line through P-naught=(3,-1,1) perpendicular to the plane 3x+5y-7z=29?

Answer 1

# { (x = 3 + lambda 3), (y = -1+lambda 5), (z = 1 -lambda 7) :}#

Explanation:

Plane `Pi->3x+5y-7z=29` can be written as

`<< vec v , p-p_0 >> = 0`

in which `p = {x,y,z}, p_0 = {x_0,y_0,y_0}` and `vec v` is a normal vector to the plane and `p-p_0` are segments pertaining to the plane.

For `Pi` we have `vec v = {3,5,-7}` so the line perpendicular to `Pi` passing by `q_0 = {3,-1,1}` is

`l->p = q_0 + lambda vec v` or

# { (x = 3 + lambda 3), (y = -1+lambda 5), (z = 1 -lambda 7) :}#