Find the equations of the tangent plane and the normal line to the given surface

Find the equations of the tangent plane and the normal line to the given surface at the given point...

$\displaystyle z=e^xcos(y), (0,0,1)$

I just need a check of my work, I'm studying for my test, and this problem isn't in the back of the book to check.

$\displaystyle z_x=e^x|(0,0) =1$

$\displaystyle z_y=-e^xsin(y)|(0,0) =0$

So plugging into the plane formula, I got

$\displaystyle z-1=1(x-0)+0(y-0) z=x+1$

And the normal vector would be

<1,0,-1> (this I'm unsure of)

Thus giving parametric line equations of...

$\displaystyle x=t, y=0, z=1-t$