About vectors.......

Let length of vector= r=√(a_x²+a_y²)
Let θ be the angle with x-axis.
a_x=rCosθ
a_y=rSinθ
a'_x =rCos(θ-α)= r(CosθCosα+SinθSinα)
= a_xCosα+a_ySinα
a'_y = rSin(θ-α)= r(SinθCosα-CosθSinα)
= a_yCosα - a_xSinα

let the initial angle of poinr P be θ....
let r be the dist of point from origin
then in new system
X=rcos(θ-α)
and in the old sys x=rcosθ

similary

Y=rsin(θ-α)

..
y=rsinθ

now substitute the values....

as tanθ=a_y/a_x