I'm an undergrad in physics (first year) and I was having a problem about analyzing when two surface functions f(x,y,z) = some constant and g(x,y,z) = some other constant, that are tangent with each other. I was thinking that if they tangent each other at a certain point, their gradients must be aligned. Usually, when I think of two vectors aligned with each other, I think about that the area of a parallelogram created from these two vectors being 0 which can be calculated from the determinant of the vectors because that has always made sense to me visually.
I then want to set the determinant between these two vectors to 0 and analyze for which points (x,y) this equality is true. This becomes a problem in this case as I have two 3D vectors. In physics, I'm used to being able to calculate the determinant between two 3D vectors if the first row is made out of the base vector (i,j,k) but that will give me an answer that doesn't make sense in this situation.
The possible solutions I'm thinking of seem too complicated for this problem which I don't feel should be necessary...
How do I think about this problem and how do I solve it?
1) You don't need 3D vectors, since the normal to the surfaces is $\mathbf n_f=(f'_x,f'_y,1)$ and $\mathbf n_g=(g'_x,g'_y,1)$, so if $\mathbf n_f$ and $\mathbf n_g$ are aligned, then $\nabla f = \nabla g$.
2) If you want to understand whether two vectors $\mathbf u$ and $\mathbf v$ are aligned in general, you can compute the cosine of angle between them as $$ \cos \alpha=\frac{\mathbf u}{|\mathbf u|}\cdot\frac{\mathbf v}{|\mathbf v|}=\frac{\mathbf u\cdot\mathbf v}{|\mathbf u||\mathbf v|}. $$ If $|\cos\alpha|=1$, then vectors are collinear.
You can use the reasoning with the area of parallelogram too. The area of the parallelogram on 2 vectors: $$ A = \sqrt{\det\begin{pmatrix}\mathbf u^2 & \mathbf u\cdot\mathbf v\\ \mathbf u\cdot\mathbf v& \mathbf v^2\end{pmatrix}} = \sqrt{\mathbf u^2\mathbf v^2 - (\mathbf u\cdot\mathbf v)^2} $$
You can easily see, that $A=0$ is an equivalent to the previous criterion.