I am trying to understand why in an exercise aiming to prove that the Identity matrix is orthogonal we use the dot product and not matrix multiplication on the columns. So if the identity matrix is of 3 rows and 3 columns we have: $$ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{matrix} $$ And then the columns which I am trying to multiply are: $ \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} $ and $ \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} $ And I see that the dot product of this two columns makes 0 which sounds correct from geometry but I was wondering why it isn't used matrix multiplication of the two columns instead? I see that both columns are of size 3x1 which I think actually means that they are not compatible for multiplication. So I guess I have two questions:
- Why are we using dot product and not matrix multiplication?
- And second is even matrix multiplication possible on two vectors of size 3x1?
Thank you in advance!
PS Pardon my MathJax syntax, never worked with this syntax before.