If r1 = 3, r2 = 4, and Δθ = π, what is the distance d between the two points in polar coordinates?

If r1 = 3, r2 = 4, and Δθ = π, what is the distance d between the two points in polar coordinates?

• A. 5
• B. 7
• C. 1
• D. 6

Answer: (B)

Distance between two polar points is found by treating the two radius vectors as sides of a triangle and applying the Law of Cosines: d^2 = r1^2 + r2^2 - 2 r1 r2 cos(Δθ), where Δθ is the difference in their angles. Here Δθ = π, so cos(π) = -1. Then d^2 = 3^2 + 4^2 - 2·3·4·(-1) = 9 + 16 + 24 = 49, giving d = 7. Geometrically, when the angle between the radius vectors is π, the points lie on opposite directions along the same line, so the distance is the sum of the radii: 3 + 4 = 7.

Distance between two polar points is found by treating the two radius vectors as sides of a triangle and applying the Law of Cosines: d^2 = r1^2 + r2^2 - 2 r1 r2 cos(Δθ), where Δθ is the difference in their angles. Here Δθ = π, so cos(π) = -1. Then d^2 = 3^2 + 4^2 - 2·3·4·(-1) = 9 + 16 + 24 = 49, giving d = 7. Geometrically, when the angle between the radius vectors is π, the points lie on opposite directions along the same line, so the distance is the sum of the radii: 3 + 4 = 7.