Step 1 :Calculate vector w: \(w = 2v - u = 2\langle 3, 4 \rangle - \langle 1, -4 \rangle = \langle 5, 12 \rangle\)
Step 2 :Find the magnitude of vector w: \(\|w\| = \sqrt{5^2 + 12^2} = 13\)
Step 3 :Find the angle between vectors v and u: \(\cos\theta = \frac{v \cdot u}{\|v\|\|u\|} = \frac{-13}{5 \times 4.12} \Rightarrow \theta = 129.09°\)
Step 4 :Find the distance between the terminal points of vectors v and u: \(d = \sqrt{(3 - 1)^2 + (4 - (-4))^2} = 8.25\)
Step 5 :\boxed{\text{Final Answer:}}
Step 6 :A. Vector w = \(\boxed{\langle 5, 12 \rangle}\)
Step 7 :B. The magnitude of vector w = \(\boxed{13}\)
Step 8 :C. The angle between vectors v and u = \(\boxed{129.09°}\)
Step 9 :D. The distance between the terminal points of vectors v and u = \(\boxed{8.25}\)