Answer
$142.1^{\circ}$
Work Step by Step
Step 1: We let $\textbf {u}=\langle 3,-2 \rangle$ and $\textbf {v}=\langle -1,3 \rangle$
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: $\cos\theta=\frac{\langle 3,-2 \rangle\cdot\langle -1,3 \rangle}{|\langle 3,-2 \rangle||\langle -1,3 \rangle|}$
Step 4: $\cos\theta=\frac{3(-1)-2(3)}{\sqrt (3^{2}+(-2)^{2})\cdot\sqrt ((-1)^{2}+3^{2})}$
Step 5: $\cos\theta=\frac{-3-6}{\sqrt (9+4)\cdot\sqrt (1+9)}$
Step 6: $\cos\theta=\frac{-9}{\sqrt (13)\cdot\sqrt (10)}$
Step 7: $\cos\theta=\frac{-9}{\sqrt (130)}$
Step 8: $\theta=\cos^{-1}(\frac{-9}{\sqrt (130)})$
Step 9: Solving using the inverse cos function on the calculator,
$\theta=\cos^{-1}(\frac{-9}{\sqrt (130)})\approx142.1^{\circ}$
The vectors are not orthogonal.