Dear Penchant,

First of all I think you have done some mistakes in your calculations.

Therefore the normal vector is

Using this result you would get the equation of the plane as,

Since,

Therefore the line coincides with the x axis.

Since (1,0,0) is on the plane which we are considering the line intersects the plane.

Hope this will help you. If you have any questions please don't hesitate to reply me.