Answer
$\textbf {u}\cdot\textbf {v}=\langle -4,7 \rangle\cdot\langle -14,-8 \rangle=-4(-14)+7(-8)=56-56=0$
Since $\textbf {u}\cdot\textbf {v}=0$, the two vectors are orthogonal.
Work Step by Step
Step 1: 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 2: $\cos\theta=\frac{\langle -4,7 \rangle\cdot\langle -14,-8 \rangle}{|\langle -4,7 \rangle||\langle -14,-8 \rangle|}$
Step 3: $\cos\theta=\frac{-4(-14)+7(-8)}{\sqrt ((-4)^{2}+7^{2})\cdot\sqrt ((-14)^{2}+(-8)^{2})}$
Step 4: $\cos\theta=\frac{56-56}{\sqrt (16+49)\cdot\sqrt (196+64)}$
Step 5: $\cos\theta=\frac{0}{\sqrt (65)\cdot\sqrt (260)}$
Step 6: $\cos\theta=0$
Step 7: $\theta=\cos^{-1}(0)$
Step 8: Solving using the inverse cos function on the calculator,
$\theta=\cos^{-1}(0)=90^{\circ}$
Since the angle between the two vectors is $90^{\circ}$, the two vectors are orthogonal.
