Decide how much you want to rotate the vector. Call this angle theta. This should be measured counterclockwise. Suppose you want to rotate the vector (3,4) 60 degrees counter clockwise, then theta = 60, x= 3 and y = 4.
Find sine of theta. Sin(60) = 0.87.
Find cosine of theta. Cos(60) = 0.5
Multiply x by cosine theta. 3*0.5 = 1.5
Multiply y by sine of theta. 4*0.87=3.38.
Add the results from Steps 4 and 5. This is the x component of the rotated vector: 3.38+1.5 = 4.68
Multiply x by sine theta and y by cosine theta and add them. This is the y component of the new vector. In the example x*sin(theta) = 3*0.87 = 2.61; y*cos(theta) = 4*0.5 = 2. 2.61 + 2 = 4.61. The rotated vector is (3.38, 4.61).