Final answer:
In a two-dimensional vector system, x and y coordinates can be set arbitrarily, but are often limited by relationships within the system. X is typically an independent variable, whereas y is a dependent variable that must be calculated. Special cases exist, such as at the origin, where restrictions may not apply.
Step-by-step explanation:
In mathematics, particularly when dealing with a two-dimensional vector system, the coordinates x and y can indeed be set arbitrarily to some extent. When you are given a system where no two vectors are parallel, you typically project those vectors onto a chosen coordinate system with a horizontal x-axis and a vertical y-axis. However, the values you can assign to x and y may be constrained by particular relationships between the two, such as the equations provided or specific rules in physical systems, such as quantum mechanics.
For instance, in the context of classical physics versus quantum mechanics, the positioning of particles such as being at x₁ or x₂ can be quite different. While classical physics presupposes a definite position, quantum mechanics allows for superposition, meaning a particle can be in multiple states or positions simultaneously.
Therefore, while x (the independent variable) can typically be chosen freely, y (the dependent variable) will often need to be calculated based on x in accordance with a given equation or system of equations. In special cases, like at the origin of the coordinate system (where x = y = 0), you might be able to set them without restriction since certain equations may not apply, or the variables are undefined.