Introduction:
When a coordinate system of x y axis is rotated by an angle theta in anticlockwise direction in the same plane, the unit vector along new set of axes changes. This is because the new set of axes are not parallel to the original x y axis.

Derivation:
Let's assume that the original x y axis is rotated by an angle theta in anticlockwise direction to form a new set of axes x' y'. The unit vector i along the original x axis can be expressed as i = (1,0) and the unit vector j along the original y axis can be expressed as j = (0,1).

Now, let's consider a point P in the original coordinate system. The coordinates of P in the original system can be expressed as (x,y). In the new coordinate system, the coordinates of P can be expressed as (x',y').

Rotation Matrix:
To find the relationship between the coordinates in the original system and the new system, we need to use a rotation matrix. The rotation matrix is given by:

|cos(theta) -sin(theta)|
|sin(theta) cos(theta)|

Transformation of i:
To find the unit vector i' along the new x' axis, we need to transform the original unit vector i using the rotation matrix. This gives us:

i' = (1,0) * |cos(theta) -sin(theta)| = (cos(theta), -sin(theta))

|sin(theta) cos(theta)|

Transformation of j:
Similarly, to find the unit vector j' along the new y' axis, we need to transform the original unit vector j using the rotation matrix. This gives us:

j' = (0,1) * |cos(theta) -sin(theta)| = (sin(theta), cos(theta))

|sin(theta) cos(theta)|

Conclusion:
Therefore, the unit vector along the new set of axes are given by i' = (cos(theta), -sin(theta)) and j' = (sin(theta), cos(theta)). These new unit vectors are not parallel to the original x y axis and are rotated by an angle theta in anticlockwise direction.