Cartesian and Polar Coordinates

Cartesian Coordinates

Cartesian Coordinates

In a system formed by a point, O, and an orthonormal basis at each point, P, there is acorresponds a vector, Vector, in the plane such that:

Cartesian Vector

The coefficients x and y of the linear combination are called coordinates of point P.

The first, x, is the abscissa.

The second, y, is the ordinate.

As the linear combination is unique, each point corresponds to a pair of numbers and a each pair of numbers to a point.


Polar Coordinates

When the length, r, and the angle, α, (it makes with the x-axis), of the vector are known, the polar coordinates of P are (r, α):


Polar-Cartesian Coordinate Conversion

Polar-Cartesian Coordinate Conversion

x = r · cos α

y = r · sen α

Examples

Convert the following polar coordinates to the cartesian system:

(2, 120º)

Coordinate Conversion Example

Coordinate Conversion

Coordinate Conversion

(1, 0º)= (1, 0)

(1, 180º) = (−1, 0)

(1, 90º) = (0, 1)

(1, 270º) = −(0, −1)


Cartesian-Polar Coordinate Conversion


Magnitude

Magnitude of a Vector Formula

Angle

Angle of a Vector Formula

Examples

Convert the following cartesian coordinates to the polar system:

Coordinate Conversion Example

Coordinate Conversion

Coordinate Conversion

(2, 60º)


Coordinate Conversion Example

Coordinate Conversion

Coordinate Conversion

(2, 120º)


Coordinate Conversion Example

Coordinate Conversion

Coordinate Conversion

(2, 240º)


Coordinate Conversion Example

Coordinate Conversion

Coordinate Conversion

(2, 300º)


(2, 0)

Coordinate Conversion

Coordinate Conversion

2


(−2, 0)

Coordinate Conversion

Coordinate Conversion

(2, 180º)


(0, 2)

Coordinate Conversion

Coordinate Conversion

(2, 90º)


(0, −2)

Coordinate Conversion

Coordinate Conversion

(2, 270º)