Given (a * b) . (c * d)

Let r = (c * d)

Now, (a * b) . (c * d) = (a * b) . r

=> (a * b) . (c * d) = a . (b * r)         {interchanging dot and cross}

=> (a * b) . (c * d) = a . {b * (c * d)}

=> (a * b) . (c * d) = a . {(b . d)c - (b . c)d}

=> (a * b) . (c * d) = (a . c)(b . d)- (a . d) (b . c)

=> (a * b) . (c * d) =  |a . c   a . d|

                                   |b . c   b . d|