Conventional notation: $\vec{a^T}\vec{a} = ||\vec{a}||$ or $\vec{a^T}\vec{a} = ||\vec{a}||^2$ Just want to make sure I grasp the conventional notation on this once and for all. If I recollect correctly the following should be correct, right?: $\vec{a}\cdot \vec{a} = \vec{a^T}\vec{a} = ||\vec{a}||^2$ But I'm not sure if I've actually seen this relationship in some places, which is in conflict with the above. So this must be wrong or am I missing something?: $\vec{a^T}\vec{a} = ||\vec{a}||$

The correct one is $$ \vec a^T \vec a = \|\vec a\|^2. $$ The intuition behind this is that the quantity on the left is quadratic with respect to $\vec a$, so the right hand side must be quadratic as well (which would not be the case for $\|\vec a\|$): Take real $t>0$, then $$ (t\vec a)^T(t\vec a) = t^2 \vec a^T \vec a = t^2 \|\vec a\|^2 = \|t \vec a\|^2 $$ holds for all $t>0$. This computation would not work with the incorrect $\vec a^T \vec a = \|\vec a\|$.