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L 4 x = L 4 cos θ 4 = ( 80.0 m ) cos ( −90 ° ) = 0 , L 4 y = L 4 sin θ 4 = ( 80.0 m ) sin ( −90 ° ) = −80.0 m, L 4 = L 4 x i ^ + L 4 y j ^ = ( −80.0 m ) j ^ .

On the last leg, the magnitude is L 5 = 150.0 m and the angle is θ 5 = −23 ° + 270 ° = + 247 ° ( 23 ° west of south), which gives

L 5 x = L 5 cos θ 5 = ( 150.0 m ) cos 247 ° = −58.6 m, L 5 y = L 5 sin θ 5 = ( 150.0 m ) sin 247 ° = −138.1 m, L 5 = L 5 x i ^ + L 5 y j ^ = ( −58.6 i ^ 138.1 j ^ ) m .

Check Your Understanding If Trooper runs 20 m west before taking a rest, what is his displacement vector?

D = ( −20 m ) j ^

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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 i ^ and 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 φ 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 r ^ . The second unit vector t ^ is a vector orthogonal to the radial direction r ^ . The positive + t ^ direction indicates how the angle φ 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 , φ ) . [link] is valid for any vector, so we can use it to express the x - and y -coordinates of vector r . In this way, we obtain the connection between the polar coordinates and rectangular coordinates of point P :

{ x = r cos φ y = r sin φ .
Vector r points from the origin of the x y coordinate system to point P. The angle between the vector r and the positive x direction is phi. X equals r cosine phi and y equals r sine phi. Extending a line in the direction of r vector past point P, a unit vector r hat is drawn in the same direction as r. A unit vector t hat is perpendicular to r hat, pointing 90 degrees counterclockwise to r hat.
Using polar coordinates, the unit vector r ^ defines the positive direction along the radius r (radial direction) and, orthogonal to it, the unit vector t ^ defines the positive direction of rotation by the angle φ .

Polar coordinates

A treasure hunter finds one silver coin at a location 20.0 m away from a dry well in the direction 20 ° north of east and finds one gold coin at a location 10.0 m away from the well in the direction 20 ° north of west. What are the polar and rectangular coordinates of these findings with respect to the well?

Strategy

The well marks the origin of the coordinate system and east is the + x -direction. We identify radial distances from the locations to the origin, which are r S = 20.0 m (for the silver coin) and r G = 10.0 m (for the gold coin). To find the angular coordinates, we convert 20 ° to radians: 20 ° = π 20 / 180 = π / 9 . We use [link] to find the x - and y -coordinates of the coins.

Solution

The angular coordinate of the silver coin is φ S = π / 9 , whereas the angular coordinate of the gold coin is φ G = π π / 9 = 8 π / 9 . Hence, the polar coordinates of the silver coin are ( r S , φ S ) = ( 20.0 m , π / 9 ) and those of the gold coin are ( r G , φ G ) = ( 10.0 m , 8 π / 9 ) . We substitute these coordinates into [link] to obtain rectangular coordinates. For the gold coin, the coordinates are

{ x G = r G cos φ G = ( 10.0 m ) cos 8 π / 9 = −9.4 m y G = r G sin φ G = ( 10.0 m ) sin 8 π / 9 = 3.4 m ( x G , y G ) = ( −9.4 m , 3.4 m ) .

For the silver coin, the coordinates are

{ x S = r S cos φ S = ( 20.0 m ) cos π / 9 = 18.9 m y S = r S sin φ S = ( 20.0 m ) sin π / 9 = 6.8 m ( x S , y S ) = ( 18.9 m , 6.8 m ) .
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Practice Key Terms 8

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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