I have a vector which is defined as $$ r(x)=\left( \begin{array}{c} a_1\\a_2\\a_3 \end{array} \right) u(x,y)+ \left( \begin{array}{c} b_1\\b_2\\b_3 \end{array} \right) D_xu(x,y)+ \left( \begin{array}{c} c_1\\c_2\\c_3 \end{array} \right) D_yu(x,y). $$ My goal is to find the euclidean norm to this equation, which I obtain as $$ |r(x)|=\sqrt{|a|^2u^2+|b|^2u_x^2+|c|^2u_y^2+ 2u u_y a \centerdot c+2u u_x a\cdot b+2u_xu_yb \cdot c} $$ The internals of the square root look suspiciously like the standard $$\left(a u+b u_x+c u_y\right){}^2 = a^2 u^2+b^2 u_x^2+c^2 u_y^2+2 a b u u_x+2 a c u u_y+2 b c u_x u_y $$ but I can't see how we can have both the dot product and the squared euclidean norm in the same equation. I can only assume there's a simple trick that I'm missing which will make this work properly?
Thanks!
The credit should go to amd, since their comment above pointed me in the right direction. $$ |\hat{r}(x,y)|^2=\hat{r}(x,y) \cdot \hat{r}(x,y) $$ and thus $$ |\hat{r}(x,y)|=\sqrt{\hat{r}(x,y) \cdot \hat{r}(x,y)} $$