6. Cartesian and polar coordinates in two dimensions





If you are looking at an aeroplane on a radar screen you could determine its position by giving its distance from you and a direction or angle, say northwest. Let us see how to do this in terms of coordinates. Imagine that the point Q represents the position of the aeroplane, and you are at the origin. The length r of the line segment OQ is calculated by applying Pythagoras’ theorem to the right angled triangle ORQ. This gives This formula for r is valid for both positive and negative values of x and y. To measure an angle or direction for Q we have to measure it starting from somewhere. By convention, all angles are measured starting from the positive OX axis, increasing in the anticlockwise direction. Now we have the following diagram, in which denotes the angle QOX and the vector r represents the position vector of the aeroplane relative to the origin, that is, r = . Elementary trigonometry and a comparison with the previous diagram show that where r is the length of OQ and is the angle QOX. The position of the point Q can now be described in two ways: either by giving its Cartesian coordinates (x,y) or by giving what are called its polar coordinates (r,), where the scalars x, y, r, are linked by the equations Thus given the polar coordinates (r,) of a point Q, we can calculate the Cartesian coordinates (x,y). Conversely, given the Cartesian coordinates (x,y), we can calculate the polar coordinates (r,), since The actual value of is obtained in practice by using the inverse cosine or sine functions on a calculator, and by knowing the quadrant in which lies. As r = = + , we obtain what is known as the polar representation of the vector r, or alternatively the polar form of r,


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