School of Mathematics and Statistics, The University of Sydney
 6. Cartesian and polar coordinates in two dimensions
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The Cartesian form of a vector

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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 = ---> OQ. If Q has Cartesian coordinates (x,y), r may be expressed in terms of i, j, x and y as follows.

  Y   S                Q(x, y)     y    r  j           x O                       X       i        R

The vector ---> OR is equal to xi. Similarly the vector ---> OS is equal to yj. Now we observe that the vector ---> OQ is the sum of ---> OR and ---> OS and therefore

    ---> r = OQ  =  xi + yj.

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 = - --> OQ = xi + yj is valid in all quadrants. For example, if Q is in the second quadrant and has Cartesian coordinates (-2, 1), then r = ---> OQ = -2i + j.

                   Y      Q( - 2,1)                    S             r      j      R     - 2i     O   i                  X

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