Saturday, February 04, 2006

Vectors in Two-Dimensional (2-D) Coordinate Systems

Cartesian and Polar Coordinates

Cartesian Coordinates (x,y): A vector, V, in a 2-D Cartesian coordinate system can be written as:
    V=(Vx,Vy)=Vxi Vyj
where Vx and Vy are referred to as the vector components and i and j are unit vectors - vectors of length one - in the x and y directions respectively.

The magnitude of V is:

    |V|=√(Vx2 Vy2)

Polar Coordinates (r,θ): Written in polar coordinates,

where r is a unit vector in the r direction (at an angle of θ counter-clockwise from the x-axis).

Relations between Cartesian and polar coordinates:

    Vr = √(Vx2 Vy2)


    Vx = |V|cos(θ)

    Vy = |V|sin(θ)

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