So this isn't really a specific homework question, it's more of a general one. What is the difference between ax and i(hat)? I thought they were the same thing. Can someone please explain the difference?
No. They are directions parallel to the axes. You express vectors as the sum of scalar components times the unit vectors so that you add and subtract the individual terms like vectors (so they automatically satisfy the parallelogram rule for adding and subtracting vectors). It makes it very convenient to handle them mathematically. In physics problems, you are often resolving vectors into their components perpendicular and tangent to surfaces. In such cases, the unit vectors perpendicular and tangent to the surfaces come in particularly handy, and they are often not parallel to the coordinate axes.Ohhh so i, j, and k aren't actually on the axes?