MHB The Orthogonality of Vectors and Matrices

[Petrus]

Hello MHB,
I wounder if I did understand correct, If we got 3 vector and they all are orthogonala, $$v_1=(x_1,y_1,z_1)$$,$$v_2=(x_2,y_2,z_2)$$,$$v_3=(x_3,y_3,z_3)$$ does that also mean that the matrix orthogonal so the invrese for the matrix is transport?

Regards,
$$|\pi\rangle$$

 

[Ackbach]

Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.

 

[Petrus]

Close, but not quite. A matrix being orthogonal means that its columns are orthonormal.

How can I check if a columns are orthonormal? If I got it correct if $$v_1,v_2,v_3$$ shall be orthonormal that means that $$v_1*v_2*v_3=0$$

Regards,
$$|\pi\rangle$$

 

[Ackbach]

Check that $v_{i} \cdot v_{j}=\delta_{ij}$. Here
$$\delta_{ij}=\begin{cases}0,\quad &i \not=j \\ 1,\quad &i=j \end{cases}$$
is the Kronecker delta.