When defining a coordinate system, does it matter if it is right- or left-handed? When you are defining a coordinate system when solving a problem, do the coordinates need to be right-handed to obtain a correct solution? I feel like the answer is no because the directions of vectors like torque should be independent of the coordinate system we choose; they will just be represented by different coordinates. Therefore, I think that a left-handed coordinate system (if chosen for convenience) would still yield correct answers. Is my thinking correct?
Does it ever become important to stick to a right-handed coordinate system in higher-level physics?
 A: Choice of coordinate system is physically arbitrary, but may be important for reasons of convention, consistency, and transparency. The universe doesn't care how you describe it, but your readers might.
With regards to right-handed vs left-handed coordinate systems, note that coordinate systems are also relationships between directions, not just labels for them. For instance, you may have learned a "right-hand rule" mnemonic for evaluating the direction of the torque vector. If you're using a left-handed coordinate  system, the mnemonic won't work; you'd need a corresponding left-hand rule for cross-products.
A: Some physical quantitites are defined as the cross-product of other (pseudo-)vectors. If you want a hassle-free experience with the direction of the cross product, you often use the right-hand rule. Is it necessary? No. Is it convenient? Often, yes.
A: The choice of a coordinate system is like asking where to start a 100m race. Of course it doesn't matter for the concept of race where you start a race. Inshort,it doesn't matter.
