I'm trying to express the following relation in indicial notation $$ |\vec{u} - \vec{v}_p| \, . $$
The only way I found out is replacing the difference above by $$ \vec{u} - \vec{v}_p = \vec{v}_r \, , $$ then one can write $$ |\vec{u} - \vec{v}_p| = |\vec{v}_r| = (v_{r,i}v_{r,i})^{1/2} \, . $$
Is there anyway to express the vector difference directly in the indicial notation?
You could always do $$|\vec{u}-\vec{v}_p|=\sqrt{\sum_{i=1}^{n}(u_i-v_{p,i})^2},$$ so that you don't need the helper vector $\vec{v}_r$. Another option is to use the Einstein summation convention, and just write $$|\vec{u}-\vec{v}_p|=\sqrt{(u_i-v_{p,i})(u_i-v_{p,i})},$$ but you'd have to make sure your readers were aware that you were invoking the convention.