The question:

Show that the curve is normal to the surface at the point (1, 1, 1).

My attempt:

I found the tangent plane for the second surface, which is:

I then noticed that the curve is at point (1, 1, 1) when t = 1. I differentiated the curve, substituted t = 1, and got:

Which is the same as the tangent plane for the second surface. I'm confused, how can it be normal to the surface if it has the same tangent? Or have I calculated it wrong?