Question 5 of 5
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22-23 Precal
Consider vectors $v=\langle 3,4\rangle$ and $u=\langle 1,-4\rangle$. Calculate:
A. Vector $w=2 v-u$.
B. The magnitude of vector $\boldsymbol{w}$.
C. The angle between vectors $\boldsymbol{v}$ and $\boldsymbol{u}$.
D. The distance between the terminal points of vectors $v$ and $\boldsymbol{u}$.

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 = \(\boxed{8.25}\)