Final answer:
The distance between points P (3, -1) and Q (-5, 6) is approximately 10.6 units. The polar coordinates of point P are approximately (3.2, -18.4°).
Step-by-step explanation:
The distance between two points in a Cartesian plane can be found using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). In this case, the coordinates of point P are (3, -1) and the coordinates of point Q are (-5, 6). Plugging these values into the distance formula, we get:
d = sqrt((-5 - 3)^2 + (6 - (-1))^2) = sqrt((-8)^2 + (7)^2) = sqrt(64 + 49) = sqrt(113) ≈ 10.6
The polar coordinates of a point can be found using the formulas r = sqrt(x^2 + y^2) and θ = arctan(y / x). For point P, the Cartesian coordinates are (3, -1), so the polar coordinates are:
r = sqrt(3^2 + (-1)^2) = sqrt(9 + 1) = sqrt(10) ≈ 3.2
θ = arctan((-1) / 3) ≈ -18.4°