How to multiply two quaternions

I have two quaternions, as an example:

    w     x     y     z
1:  0.98  0.08  0.17  -0.01
2:  0.70  0.70  0.0   0.0

I need to multiply them, to get a third one, with all rotations in it.

How do I do that in PHP/C++/PAWN?

Solution

You should choose a language. In C++, the Boost.Math library includes quaternions; I don't know about the other languages you mention. Or for simple multiplication, you could just use the multiplication table (which I copied from Wikipedia):

*| 1  i  j  k
-------------
1| 1  i  j  k
i| i -1  k -j
j| j -k -1  i
k| k  j -i -1

For example, i*j gives the value in row i and column j, which is k.

So, assuming your quaternions represent w*1 + x*i + y*j + z*k, multiplication would be something like

quaternion operator*(quaternion a, quaternion b) {
    return {
        a.w * b.w - a.x * b.x - a.y * b.y - a.z * b.z,  // 1
        a.w * b.x + a.x * b.w + a.y * b.z - a.z * b.y,  // i
        a.w * b.y - a.x * b.z + a.y * b.w + a.z * b.x,  // j
        a.w * b.z + a.x * b.y - a.y * b.x + a.z * b.w   // k
    };
}

(NOTE: that's untested, and probably riddled with typos).