Verified answer

|v · w| = |v||w|cos(θ) = -4

|v x w| = |v||w|sin(θ) = 8

tan(θ) = |(v x w)|/|(v · w)| = -2

The gist of the talk obtainable (sorry, there are too many pages to supply all of the links) is that wands help to concentration the magic resident in the witch or wizard, yet wandless magic continues to be achieveable, extremely for terribly gifted people like Snape and Dumbledore. some examples ... Apparition Assuming one's Animagus form (or Metamorphpagus) Accio (the Summoning charm) Elves can do magic without utilising wands Lumos