About what angle separates the two vectors?

A good answer might be:

After rotating them figure a few times, I guessed 50°. But this is only a guess.

Applying the Formula

Lets see how close my guess is:

  1. The two vectors are represented by:
    2. f  =  (4, 3, 2)T
    g  =  (-1, 4, 4)T
  2. The lengths are:
    3. | f |2  =  (4, 3, 2)T · (4, 3, 2)T  =  16 + 9 + 4  =  29
    | g |2  =  (-1, 4, 4)T · (-1, 4, 4)T  =  1 + 16 + 16  =  33
  3. The normalized vectors are:
    4. fu  =   (4, 3, 2)T / 29
    gu  =   (-1, 4, 4)T / 33
  4. The dot product is:
    5. fu · gu  =   (4, 3, 2)T · (-1, 4, 4)T / (29 33)
      =   (-4 + 12 + 8)/(29 33)   =   16 / (29 33)   =   0.51721
  5. The angle is:
    6. cos θ =  0.51721
    θ =  arc cos 0.51721  = 58.855°

Not too far off from my guess.

QUESTION 7:

Do two vectors need to be touching at their tails for there to be an angle between them?