What's the insight for a 3x3 matrix with orthogonormal columns,the rows are also orthogonormal?

Question

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.

Answer 1

A square real matrix $M$ has columns ON if and only if $M^{-1}=^tM$ if and only if $(^tM)^{-1}=M=^t(^tM)$ if and only if $^tM$ columns are ON if and only if $M$ lines are ON.

ON = orthonormal.

Answer 2

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.