DirectX 3D rotate eyePt around lookAt

Question

I'm really stuggling with the maths i need for this maths has never been my strong point once you get into Calculus and Geometry.

So how do i rotate a vector3 around another vector3

D3DXVECTOR3 lookAt = this->cam->getLookAt();
D3DXVECTOR3 eyePt = this->cam->getEyePt();

I need to rotate lookAt around eyePt how do I do this I know I need a Matrix but what I'm supposed to fill it with and how I do the rotation in it I just don't understand.

So if some one could provide code with explanations of the steps used to do it would really help

Another note is that i only want to translate on X,Z axis as i'm rotation around Y axis so here is an image of what im trying to do

Answer 1

Take the unit vector eyePt, which will be the axis of rotations. (I presume it's a unit vector; if not, I can show you how to turn it into a unit vector.) Let's call it E:

    Ex
E = Ey
    Ez

(This is the vector (E_x, E_y, E_z), but it's hard to do mathematical notation here.)

Now construct three matrices. The identity matrix I:

    1 0 0
I = 0 1 0
    0 0 1

the tensor product of E and E, which we'll call T:

      0 -Ez  Ey
T =  Ez   0 -Ex
    -Ey  Ex   0

and the cross-product matrix of E, which we'll call P:

    ExEx ExEy ExEz
P = ExEy EyEy EyEz
    ExEz EyEz EzEz

Now choose an angle of rotation, theta (in radians). The rotation matrix will be:

R = cos(theta)I + (1-cos(theta))T + sin(theta)P

Now to rotate the vector v (which in this case is lookAt), just multiply R by it:

v_after = Rv_before