push_back appends one element; the vector's amortised cost is O(1) per insertion because capacity grows geometrically (typically doubling). nums[i] is unchecked: passing an out-of-range index is undefined behaviour, while nums.at(i) throws. Range-for (for (auto x : v)) calls begin()/end() under the hood and is the idiomatic loop in modern C++. Always take by const auto& for non-trivial element types so you avoid an unwanted copy each iteration.

emplace_back constructs the element directly inside the vector's storage from the forwarded arguments, skipping the temporary that push_back(T(args...)) would otherwise create. For a trivial type like Point, the difference is negligible because the compiler can elide the move; for types with non-trivial copy or move constructors (a std::string field, a unique_ptr member), it matters. The mental model: emplace_back is an in-place factory call; push_back is an append of an already-built object. Default to emplace_back for new code unless you specifically need a copy of an existing instance.

Without reserve, the vector reallocates every time it doubles past its current capacity, copying or moving every element each time. reserve(n) allocates capacity for n elements up front when you know (or can estimate) the final size. The size stays at zero until you actually push elements; capacity() reports the allocation. This is one of the cheapest performance wins in C++: a tight loop building a known-size vector goes from O(n log n) total work to O(n) just by adding a reserve.