Question

Two points in the Cartesian plane are A(2.00 m, −4.00 m) and B(−3.00 m, 3.00 m). Find the distance between them and their polar coordinates.

a) Distance: 7.810 m, A: (4.472 m, -63.43°), B: (5.000 m, 143.13°)
b) Distance: 5.000 m, A: (2.236 m, -63.43°), B: (3.000 m, 143.13°)
c) Distance: 6.403 m, A: (3.000 m, -37.00°), B: (4.000 m, 127.00°)
d) Distance: 6.708 m, A: (3.606 m, -53.13°), B: (4.472 m, 126.87°)

Answer 1

The distance between points A and B is 8.60 m. The polar coordinates for point A are (2√5 m, -63.43°) and for point B are (3√2 m, 143.13°).

Step-by-step explanation:

To find the distance between points A and B, we can use the distance formula in the Cartesian plane. The formula is D = sqrt((x2 - x1)^2 + (y2 - y1)^2). Plugging in the coordinates of A(2.00 m, -4.00 m) and B(-3.00 m, 3.00 m), we get D = sqrt(((-3.00) - 2.00)^2 + (3.00 - (-4.00))^2) = sqrt(25 + 49) = sqrt(74) = 8.60 m.

To find the polar coordinates of a point, we need to determine its distance from the origin (r) and its angle with respect to the positive x-axis (θ). In this case, the coordinates of point A(2.00 m, -4.00 m) can be represented as A(r, θ). Using the formulas r = sqrt(x^2 + y^2) and θ = tan^-1(y/x), we can calculate r and θ for point A. Plugging in the values, we get r = sqrt((2.00)^2 + (-4.00)^2) = sqrt(4 + 16) = sqrt(20) = 2√5 m and θ = tan^-1((-4.00)/(2.00)) = tan^-1(-2) = -63.43°.

The polar coordinates for point A are (2√5 m, -63.43°). Similarly, plugging in the values for point B(-3.00 m, 3.00 m), we get r = sqrt((-3.00)^2 + (3.00)^2) = sqrt(9 + 9) = sqrt(18) = 3√2 m and θ = tan^-1((3.00)/(-3.00)) = tan^-1(-1) = 143.13°. The polar coordinates for point B are (3√2 m, 143.13°).