Rotate vector

Rotate a vector by an angle

A vector (x, y) with angle A to the x axis

Rotate (x, y) by angle B to (x', y'), so the angle between vector(x', y') and the x axis is (A + B)

|

|    (x', y')

|   /

|  /

| /B     (x, y)

|/_A______________

Let the length of the vector be R 

x = R * cosA

y = R * sinA

so cosA = x/R and sinA = y/R

x' = R * cos(A+B) = R * (cosA * cosB - sinA * sinB)

y' = R * sin(A + B) = R * (sinA * cosB + cosA * sinB)

Replace the cosA and sinA in, we get

x' = R * (x/R * cosB - y/R * sinB)

y' = R * (y/R * cosB + x/R * sinB)

i.e.

x' = xcosB - ysinB

y' = xsinB + ycosB

[[x'],[y']] = [[cosB, -sinB], [sinB, cosB]] <dot> [[x],[y]]

In python using numpy, define the rotation matrix as

mat = [[cosB, -sinB], [sinB, cosB]]

v1 = [[x],[y]]

then the v2 after rotation 

v2 = np.dot(mat, v1)