Do gradients of level curves at tangent point point at same direction?

Question

I watched the Lagrange multipliers video here and it was mentioned in minute 2:50 that the gradients of both level curves at tangent point point at the same direction

Is this guaranted that they will point always at the same direction? If it is, can you please explain and/or provide a link to an article with an easy explanation?

Answer 1

Good catch! The video narrator is certainly wrong when he said that "they're pointing in the same direction." Note that up to that point he was only saying that the two gradients would be proportional, which is absolutely correct. Equivalently, being proportional means that they are parallel. But they do NOT have to point in the same direction. In fact, it doesn't matter at all whether they point in the same or in opposite directions — it only matters that they are parallel (proportional).

To give them the benefit of the doubt, we can presume that he knows the stuff, and just misspoke at that point. Still, it is very unfortunate that an educational resource would make such a blatant mistake, confusing students who are trying to learn from them…