How to Find the Cartesian Equation for a Plane Parallel to the x-axis?

[aeromat]

Homework Statement

Determine the Cartesian equation for the plane with the following:
1) through the points (1,2,1) and (2,1,4)
2) parallel to x-axis

Homework Equations

Parametric Equations
Vector Eqs
Plane Eqs

The Attempt at a Solution

I basically understand how perpendicular works, but not sure how parallel will change this situation. I know that I will get a direction vector out of this from the first two points, which will be (1,-1,3), but then I don't know what to do after..

 

[LCKurtz]

OK, you have so far that D = <1,-1,3> is in the plane, so it is perpendicular to the plane's normal, call it N. If (x,y,z) is a variable point in the plane you can make another vector in the plane and perpendicular to the plane's normal N.

So the problem boils down to what does the normal vector N to the plane look like if the plane is parallel to the x axis? What can you say about its components?

 

[HallsofIvy]

If a plane is parallel to the x-axis, then the vector \vec{i}, or (1, 0, 0) in your notation, is in the plane. Use the two given points to find another vector in the plane and take their cross product to get a vector normal to the plane.

Another way to look at it is this: if the plane is parallel to the x-axis, there will be NO x in the formula. Write Ay+ Bz= 1 and use the two points to get two equations to solve for A and B.