Undergrad Gradient vectors and level surfaces

[Haku]

TL;DR

I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that point? Or is there some other relationship between the tangent vector and level surface?

Homework Statement:: Wondering about the relationship between gradient vectors, level surfaces and tangent planes
Relevant Equations:: .

I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that point? Or is there some other relationship between the tangent vector and level surface?

 

[Haku]

I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that point? Or is there some other relationship between the tangent vector and level surface?

 

[Mark44]

Homework Statement:: Wondering about the relationship between gradient vectors, level surfaces and tangent planes
Relevant Equations:: no equations

I know that the gradient vector is orthogonal to the level surface at some point p, but is the gradient vector also orthogonal to the tangent vector at that point?

Yes.
Consider $w = f(x(t), y(t), z(t))$.
If w is held constant, you get a level surface.
Differentiation with respect to t yields $\frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt} = 0$
The above can be rewritten as $\nabla f \cdot \vec{\dot x} = 0$, which shows that the gradient of f is orthogonal to the tangent vector. Here $\vec{\dot x}$ is the vector $(\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt})$.

Or is there some other relationship between the tangent vector and level surface?

 

[anuttarasammyak]

The tangent vector in in the level surface.