Question

Two points in a plane have polar coordinates P1(2.500m,π/6) and P2(3.800m,2π/3).

What are the Cartesian coordinates of the two points and the distance between them in the Cartesian coordinate system?
a) P1: (1.30 m, 2.27 m), P2: (1.30 m, 2.27 m), Distance: 0.00 m
b) P1: (1.30 m, 2.27 m), P2: (−2.08 m, 3.60 m), Distance: 4.00 m
c) P1: (1.30 m, 2.27 m), P2: (−1.30 m, −2.27 m), Distance: 5.00 m
d) P1: (2.08 m, 3.60 m), P2: (−2.08 m, 3.60 m), Distance: 0.00 m

Answer 1

To convert polar coordinates to Cartesian coordinates, we use the cosine and sine functions related to the angle and radial distance. For P1(2.500m, π/6), the Cartesian coordinates are approximately (2.165 m, 1.250 m), while for P2(3.800m, 2π/3), they are approximately (-1.900 m, 3.290 m). The distance between these points is roughly 4.47 m.

Step-by-step explanation:

To convert polar coordinates to Cartesian coordinates, we use the relationships x = r • cos(φ) and y = r • sin(φ), where r is the radial distance and φ is the angle with the positive x-axis. For the first point P1(2.500m, π/6), the Cartesian coordinates are calculated as:

• x = 2.500 • cos(π/6) = 2.500 • (√3/2) = 2.165 m (approximately)
• y = 2.500 • sin(π/6) = 2.500 • (1/2) = 1.250 m (approximately)

For the second point P2(3.800m, 2π/3), the Cartesian coordinates are:

• x = 3.800 • cos(2π/3) = 3.800 • (-1/2) = -1.900 m (approximately)
• y = 3.800 • sin(2π/3) = 3.800 • (√3/2) = 3.290 m (approximately)

To find the distance between these two points, we use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2) = √((-1.900 - 2.165)^2 + (3.290 - 1.250)^2) = √((4.065)^2 + (2.040)^2) = 4.472 m (approximately), or 447.2 cm when rounded to the nearest centimeter.