The normal plane is the plane at a point P on the curve C that consists of all lines that are orthogonal to the tangent vector. Because of that property, the tangent vector is a perfect vector to chose as the "normal vector" of our normal plane. . . Hopefully you got all that.
Therefore, to find the tangent vector you simply take the derivative of your curve C, and then plug in your point P:
We know the normal vector to our normal plane must be parallel to the plane 6x+6y-8z=1, which means our normal vector should be <6, 6, -8>.
See if you can take where it from here.