Surface patch that is an isometry proof

Question

Question:

Let $σ: R2 ->S$ be a surface patch that is also an isometry. Prove that $σ_u$ and $σ_v$ are perpendicular.

So essentially for this proof I'm thinking I want to relate the dot product to the fact that if

$σ_u . σ_v=0$

Then that means they are perpendicular

So Let's say $σ_u=(u_1,u_2)$ and $σ_v=(v_1,v_2)$

Then $σ_u . σ_v=(u_1,u_2).(u_1,v_2)=u_1v_1 +u_2v_2$

But how can I prove this is $0$? Could I use other properties of $σ_u$ and $σ_v$? Perhaps length?

I'm not even sure if my premise or approach to this question is correct.