This looks very similar to Lagrange's Identity. Which states:
For 3-space, you could use
Expand out and confirm the equality.
may be written as:
OK, so I have a proof which I have am having a hard time prooving. Well here it is |a cross b| = ((a dot a) (b dot b) - (a dot b)sqrd) to the power of 1/2
(Also, it seems i have posted in the wrong section, if moving the thread to the Euclidean Geometry section is at all possible please let me know