Given the separation vector and its magnitude:
$\displaystyle \vec{r} =( x-x') \ \hat{i} +( y-y') \ \hat{j} +( z-z') \ \hat{k}$
$\displaystyle r=\left[( x-x')^{2} +( y-y')^{2} \ +( z-z')^{2}\right]^{1/2} \ $
and a new substitution being:
$\displaystyle \vec{\alpha} =( x-x')^{2} \ \hat{i} +( y-y')^{2} \ \hat{j} +( z-z')^{2} \ \hat{k}$
I am unable to prove that the following statement is true:
$\displaystyle \alpha ^{3/2} =r^{2}$