Answer
$135^{\circ}$
Work Step by Step
Step 1: We let $\textbf {u}=\langle 2,1 \rangle$ and $\textbf {v}=\langle -3,1 \rangle$
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: $\cos\theta=\frac{\langle 2,1 \rangle\cdot\langle -3,1 \rangle}{|\langle 2,1 \rangle||\langle -3,1 \rangle|}$
Step 4: $\cos\theta=\frac{2(-3)+1(1)}{\sqrt (2^{2}+1^{2})\cdot\sqrt ((-3)^{2}+1^{2})}$
Step 5: $\cos\theta=\frac{-6+1}{\sqrt (4+1)\cdot\sqrt (9+1)}$
Step 6: $\cos\theta=\frac{-5}{\sqrt (5)\cdot\sqrt (10)}$
Step 7: $\cos\theta=\frac{-5}{\sqrt (5)\sqrt (2)\sqrt (5)}$
Step 8: $\cos\theta=\frac{-5}{5\sqrt 2}$
Step 9: $\cos\theta=\frac{-1}{\sqrt 2}$
Step 10: $\theta=\cos^{-1}(\frac{-1}{\sqrt 2})$
Step 11: Solving using the inverse cos function on the calculator,
$\theta=\cos^{-1}(\frac{-1}{\sqrt 2})=135^{\circ}$