6. Cartesian and polar coordinates in two dimensions





After defining a coordinate system XOY we introduce two basic vectors, namely a vector of length one in the direction of the positive OX axis, and a vector of length one in the direction of the positive OY axis. By convention, these unit vectors are called called i and j, respectively. Let r be any vector parallel to the XOY plane. Then by translating r so that its tail is at the point O, there is a unique point Q in the XOY plane such that r = . If Q has Cartesian coordinates (x,y), r may be expressed in terms of i, j, x and y as follows. The vector is equal to xi. Similarly the vector is equal to yj. Now we observe that the vector is the sum of and and therefore This formula, which expresses r in terms of i, j, x and y, is called the Cartesian representation of the vector r in two dimensions. We say x is the component of r along the OX axis and y is the component of r along the OY axis. The formula r = = xi + yj is valid in all quadrants. For example, if Q is in the second quadrant and has Cartesian coordinates (2, 1), then r = = 2i + j.


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