Tuesday, November 22, 2011

Are vector components always considered from the Cartesian origin?

I'm learning vector algebra and I'm being told that you can add or subtract vectors by adding their 'components', but unlike with a line segment only a single x and y value are given for each vector. These two things lead me to believe that when this calculation is performed, the point of the vector that does not have an arrow in it must be the origin of the Cartesian plane. Is this correct?|||Actually there are all kinds of coordinate system. Each of which has different ways to break a vector into its component. The Cartesian you know about. But two other popular coordinate systems are spherical and cylindrical.





The three dimensions in a spherical system are omega, rho, and R. Omega is an angle in the horizontal plane, rho is an angle in the vertical plane, and R is a radius. Typically these three dimensions map as x = R cos(rho)cos(omega), y = R cos(rho)sin(omega), and z = R sin(rho).





The cylindrical coordinates also map onto the x,y,z of Cartesian coordinates. But, as you can see, the spherical and cylindrical coordinate systems do have their counterpart in the Cartesian system. That is to say, even when a vector is given in one of the other coordinate systems, its components can be converted into Cartesian.





Vectors can start and terminate anywhere within a coordinate system. They do not have to begin or end with the origin. But that does mean extra work.





So to ease the amount of calculations needed to come up with the components when the vector is off origin, we simply move it over to the origin mathematically and work the problem that way. Moving the vector to the origin is called translation. Once the result is obtained for the translated vector at the origin, we move the answer back to the initial set of coordinates and that would be the answer of the problem.

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