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In finding the hill's highest point, students need to set the gradient of h equal to zero. Several of my students had trouble figuring out what to do with this (a vector set equal to zero), and did not realize that the vector equation gave us a separate equation for each component. It seemed to help when I wrote zero as $0= 0 \hat{i} + 0 \hat{j} $.

I think there is a real issue of recognizing that a vector equation is actually an equation in each component. But I also think students sometimes “forget” how to do things they know, in unfamiliar situations. They certainly know how to solve a simple system of 2 linear equations, but they forget that when they are up to their noses in gradients and other strange concepts.