in planes, inner product = 0

Plain equations are expressed as f(x,y,z) = ax + by + cz + d = 0.

a=b=1, d=0 ⇒ a + b + cz = 0, c≠0 ⇔ z = (-a-b)/c

∇f = (a,b,c)

∇f・(1,1,(-a-b)/c) = 0

Therefore ∇f ⊥(1,1,(-a-b)/c).

From another point of view,

df = ∂f/∂x dx + ∂f/∂y dy + ∂f/∂z dz = 0 ⇔ ∇f・(dx,dy,dz) = 0 ⇔ ∇f⊥(dx,dy,dz)