Answer
The pair of vectors is not orthogonal.
Work Step by Step
Step 1: According to vector notation, $\textbf {u}=\langle \sqrt 5,-2 \rangle$ and $\textbf {v}=\langle -5,2\sqrt 5 \rangle$
Step 2: We substitute vectors $\textbf {u}$ and $\textbf {v}$ in the formula for finding the angle between a pair of vectors, $\cos\theta=\frac{\textbf {u}\cdot\textbf {v}}{|\textbf {u}||\textbf {v}|}$
Step 3: $\cos\theta=\frac{\langle \sqrt 5,-2 \rangle\cdot\langle -5,2\sqrt 5 \rangle}{|\langle \sqrt 5,-2 \rangle||\langle -5,2\sqrt 5 \rangle|}$
Step 4: $\cos\theta=\frac{\sqrt 5(-5)-2(2\sqrt 5)}{\sqrt ((\sqrt 5)^{2}+(-2)^{2})\cdot\sqrt ((-5)^{2}+(2\sqrt 5)^{2})}$
Step 5: $\cos\theta=\frac{-5\sqrt 5-4\sqrt 5}{\sqrt (5+4)\cdot\sqrt (25+20)}$
Step 6: $\cos\theta=\frac{-9\sqrt 5}{\sqrt (9)\cdot\sqrt (45)}$
Step 7: $\cos\theta=\frac{-9\sqrt 5}{\sqrt (9)\cdot\sqrt (9\times5)}$
Step 8: $\cos\theta=\frac{-9\sqrt 5}{3\times3\times\sqrt 5}$
Step 9: $\cos\theta=\frac{-9\sqrt 5}{9\sqrt 5}=-1$
Step 10: $\theta=\cos^{-1}(-1)$
Step 11: Solving using the inverse cos function on the calculator,
$\theta=\cos^{-1}(-1)=180^{\circ}$
Since the angle between the two vectors is not $90^{\circ}$, the two vectors are not orthogonal.