Question

Two points in the xy plane have Cartesian coordinates (5.50, −4.00) m and (−6.00, 6.00) m.
(a) Determine the distance between these points.
m

(b) Determine their polar coordinates.
(5.50, −4.00) r = m
(5.50, −4.00) θ = ° counterclockwise from the +x-axis
(−6.00, 6.00) r = m
(−6.00, 6.00) θ = ° counterclockwise from the +x-axis

Answer 1

$$A(5.50, -4.00),\quad B(-6.00, 6.00)$$

(a) The distance between points

$$d=\sqrt{(x_A-x_B)^2+(y_A-y_B)^2}$$

$$=\sqrt{(5.50+6.00)^2+(-4.00-6.00)^2}=15.2\:\rm m$$

(b) The polar coordinates

$$r=\sqrt{x^2+y^2}, \quad \theta=\tan^{-1}\frac{y}{x}$$ $$r_A=\sqrt{(5.50)^2+(-4.00)^2}=6.80,$$

$$\theta_A=\tan^{-1}\left(\frac{-4.00}{5.50}\right)=324^{\circ},$$ $$r_B=\sqrt{(-6.00)^2+(6.00)^2}=8.49,$$

$$\theta_B=\tan^{-1}\left(\frac{6.00}{-6.00}\right)=135^{\circ}$$

So

$$A(6.80, 324^{\circ}),\quad B(8.49, 135^{\circ})$$