Minimum Scalar Product
Jelly, 6 bytes

ṢṚ×Ṣ}S


Try it online!

Using brute force is equally short:

Œ!×S€Ṃ


How it works

ṢṚ×Ṣ}S  Main link. Arguments: u (vector), v (vector)

Ṣ       Sort the components of u.
 Ṛ      Reverse.
   Ṣ}   Sort the components of v.
  ×     Multiply the results, element by element.
     S  Compute the sum of the products.