2.2 Coordinate systems and components of a vector  (Page 5/15)

On the last leg, the magnitude is $L_5=150.0\text{m}$ and the angle is ${\theta }_5=\mathrm{-23}\text{°}+270\text{°}=+247\text{°}$ $(23\text{°}$ west of south), which gives

Polar coordinates

To describe locations of points or vectors in a plane, we need two orthogonal directions. In the Cartesian coordinate system these directions are given by unit vectors $\widehat{i}$ and $\widehat{j}$ along the x -axis and the y -axis, respectively. The Cartesian coordinate system is very convenient to use in describing displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. When describing rotation, we usually work in the polar coordinate system    .

In the polar coordinate system, the location of point P in a plane is given by two polar coordinates    ( [link] ). The first polar coordinate is the radial coordinate     r , which is the distance of point P from the origin. The second polar coordinate is an angle $\phi$ that the radial vector makes with some chosen direction, usually the positive x -direction. In polar coordinates, angles are measured in radians, or rads. The radial vector is attached at the origin and points away from the origin to point P. This radial direction is described by a unit radial vector $\widehat{r}$ . The second unit vector $\widehat{t}$ is a vector orthogonal to the radial direction $\widehat{r}$ . The positive $+\widehat{t}$ direction indicates how the angle $\phi$ changes in the counterclockwise direction. In this way, a point P that has coordinates ( x , y ) in the rectangular system can be described equivalently in the polar coordinate system by the two polar coordinates $(r,\phi )$ . [link] is valid for any vector, so we can use it to express the x - and y -coordinates of vector $\overrightarrow{r}$ . In this way, we obtain the connection between the polar coordinates and rectangular coordinates of point P :