Final answer:
In Mathematics, the signs of ratios in different quadrants of the Cartesian coordinate system depend on the signs of the x and y coordinates. In Quadrant I, both coordinates are positive, leading to a positive ratio; in Quadrant II and IV, one coordinate is negative, resulting in a negative ratio; and in Quadrant III, both coordinates are negative, which makes the ratio positive.
Step-by-step explanation:
The subject of this question is the sign of ratios in different quadrants of the two-dimensional Cartesian coordinate system, which is a concept in Mathematics, specifically relating to the signs of x and y coordinates in each of these quadrants. In Quadrant I, both x and y are positive, hence any ratio (x/y, y/x) will be positive. In Quadrant II, x is negative and y is positive, so ratios will be negative. In Quadrant III, both x and y are negative, but the ratio of two negative numbers is positive. Lastly, in Quadrant IV, x is positive and y is negative, which will result in a negative ratio. Remember that for vectors, direction angles in the second and third quadrants will have an additional 180° to account for the different axis sign.
When reviewing physical problems or equations in two-dimensional graphing, you may encounter scenarios where you evaluate an expression for both the positive and negative signs, allowing for solutions where x can be positive or negative. However, in some situations, especially in physics, only the positive solution has physical significance. Furthermore, when dealing with powers and dividing by a variable, a negative exponent means the variable is in the denominator, which is relevant for simplifying equations.
Mathematical concepts such as these are fundamental for solving a wide range of problems in algebra, physics, and engineering, where Quadratic equations and Two-Dimensional (x-y) Graphing are common.