I'm programming a physics game. It seems I can use 2 systems for storing a character's movement data:
A) x & y components (Cartesian coordinates)
B) speed and direction components (polar coordinates)
It seems I need to ultimately decide on one of these 2 systems because:
A) They both represent the same information about a vector
B) It seems redundant and inefficient to maintain both
Most game programming resources I've found use Cartesian. To my understanding, all transformations like friction, rotation, acceleration, etc are combined into each vector via multiplication, division, etc. But to me, polar feels more modular and, therefore, more malleable because each vector is comprised of and can be broken down into its two elements (direction and magnitude). If I want to modify one of these independently, I can set its value without needing to deconstruct it into separate parts.
I'm guessing that different models are suitable for different types of games. But...
What trade-offs affect the decision to use Cartesian versus polar?
When does one model become cumbersome or verbose?
Or am I way off?
The premise of your question is a bit odd. Magnitude plus angle and sum of 2 basis components are both ways to specify a vector in 2-space. In either case, you record 2 scalars (i.e. you do not have a separate variable to represent the x unit vector). The choice of rectangular vs polar coordinates doesn't change the nature of something from a vector to a scalar or vice versa.
However, different representations certainly have their uses. As you mention, breaking down into orthogonal components has a ready advantage for addition of two vectors and other operations. In addition, most displays use a x-y coordinate system, so rendering is easier because you don't have to do a coordinate transform.
If your game was based on a polar coordinate system (say a ship that always faces the center of a circle), you might actually want to represent it using polar coordinates. Other than that, rectangular coordinates are generally easier to use.
Either way, sin and cos will probably become your friend. Just remember that most graphical coordinate systems have y-down as positive.