Question

Find polar coordinates (r,θ) of the point where r<0:

a) (r,θ) where r>0
b) (∣r∣,θ) where r<0
c) (−r,θ) where r<0
d) (r,π+θ) where r<0

Answer 1

Polar coordinates are a system for describing the location of a point in a plane using a radial distance (r) and an angle (θ). The polar coordinates of a point where r < 0 can be found using different scenarios. The answer provides step-by-step explanations for each scenario with relevant examples.

Step-by-step explanation:

Polar coordinates are a system for describing the location of a point in a plane using a radial distance (r) and an angle (θ). In this question, we are given that r<0. We need to find the polar coordinates in various scenarios.

a) When r>0, the polar coordinates would be (r, θ). The negative value of r indicates that the point is located in the opposite direction of the origin along the radial axis.

b) When r<0, the absolute value of r (∣r∣) represents the distance of the point from the origin. So, the polar coordinates would be (∣r∣, θ), where θ represents the angle.

c) Similarly, if we take (−r, θ), the coordinates would be negative along the radial axis, indicating the location in the opposite direction from the origin.

d) Finally, if we consider (r, π+θ), the angle π+θ represents a rotation of π radians from the initial angle θ. This means the point would be located in the opposite direction along the radial axis.