The University of Sydney - School of Mathematics and Statistics

The Cartesian form of a vector	Page 2 of 4

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.