I have made 2 vector fields on the surface of the sphere:
$\nabla f \times (0, 0, 1)$ and $\nabla f \times(1,0,0)$
Where $f(x,y,z) = x^2 + y^2 +z^2$.
In other words this is just the cross product of the normal with one of the basis axis.
When I draw a picture of this it very much looks like the vectors generated from the first vector field are orthogonal to those of the second at the same point. But I am having a really hard time showing that they are indeed orthogonal.
The Lagrange identity in $\Bbb R^3$ says that $$\langle v_1\times v_2, w_1\times w_2\rangle = \begin{vmatrix} \langle v_1,w_1\rangle & \langle v_1, w_2\rangle \\ \langle v_2, w_1\rangle & \langle w_1,w_2\rangle \end{vmatrix}.$$Then $$\langle \nabla f \times (0,0,1), \nabla f \times (1,0,0)\rangle = \begin{vmatrix} \langle \nabla f, \nabla f\rangle & \langle \nabla f, (1,0,0)\rangle \\ \langle (0,0,1),\nabla f\rangle & 0 \end{vmatrix} = -\frac{\partial f}{\partial x} \frac{\partial f}{\partial z} = -4xz.$$The problem is that the fields $(0,0,1)$ and $(1,0,0)$ are not tangent to the sphere at all points.