Question

Two points in the $xy$ plane have Cartesian coordinates $(2.00,-4.00)m$ and $(-3.00,3.00)m$. Determine (a) the distance between these points and (b) their polar coordinates.

Answer 1

a) The distance between the points is given by

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

$=\sqrt{(2.00-[-3.00]{)}^2+(-4.00-3.00{)}^2}$

$d=\sqrt{25.0+49.0}=8.60m$

(b) To find the polar coordinates of each point, we measure the radial distance to that point and the angle it makes with the $+x$ axis.

$r_1=\sqrt{(2.00{)}^2+(-4.00{)}^2}=\sqrt{20.0}=4.47m$

${\theta }_1={\tan }^{-1}(-\frac{4.00}{2.00})=-{63.4}^0$

$r_2=\sqrt{(-3.00{)}^2+(3.00{)}^2}=\sqrt{18.0}=4.24m$

${\theta }_2={135}^0$ measured from the $+x$ axis.