Answer
$45^{\circ}$
Work Step by Step
Step 1: We let $\textbf {u}=\langle 4,0 \rangle$ and $\textbf {v}=\langle 2,2 \rangle$
Step 2: The formula for finding the angle between a
pair of vectors is $\cos\theta=\frac{\textbf {u}\cdot\textbf {v}}{|\textbf {u}||\textbf {v}|}$
Step 3: $\cos\theta=\frac{\langle 4,0 \rangle\cdot\langle 2,2 \rangle}{|\langle 4,0 \rangle||\langle 2,2 \rangle|}$
Step 4: $\cos\theta=\frac{4(2)+0(2)}{\sqrt (4^{2}+0^{2})\cdot\sqrt (2^{2}+2^{2})}$
Step 5: $\cos\theta=\frac{8+0}{\sqrt (16+0)\cdot\sqrt (4+4)}$
Step 6: $\cos\theta=\frac{8}{\sqrt (16)\cdot\sqrt (8)}$
Step 7: $\cos\theta=\frac{8}{4\times\sqrt (4\times2)}$
Step 8: $\cos\theta=\frac{8}{4\times\sqrt 4\times\sqrt 2}$
Step 9: $\cos\theta=\frac{8}{8\sqrt 2}$
Step 10: $\cos\theta=\frac{1}{\sqrt 2}$
Step 11: $\theta=\cos^{-1}(\frac{1}{\sqrt 2})$
Step 12: Solving using the inverse cos function on the calculator,
$\theta=\cos^{-1}(\frac{1}{\sqrt 2})=45^{\circ}$