What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal? I know this can be easily proved with simple matrix tricks,
But I don't know the insight for this, and just feels it amazing that if I pick up 3 orthogonormal vectors in 3d space, their corresponding x,y,z portions automatically forms orthogonormal basis,too!
I've been googling a lot with no satisfactory answer, hope I can find it here,
Thanks.

Think about it this way:
Imagine the simplest case, where the three vectors are:
$$(0,0,1), (0,1,0), (0,0,1)$$
Clearly, if you take the transpose of these vector's matrix and construct new vectors, you get the same ones, which are of course orthogonal. Now, take any three orthogonal vectors - using a rotation, we can always get back to the three unit vectors.
