Using the formula, we have

[latex]\begin{align}\boldsymbol{v}\cdot \boldsymbol{u}&=\langle 5,12\rangle \cdot \langle -3,4\rangle \\ &=5\cdot \left(-3\right)+12\cdot 4 \\ &=-15+48 \\ &=33 \end{align}[/latex]

Finding the dot product, we multiply corresponding components.

[latex]\begin{align}{\boldsymbol{v}}_{1}\cdot {\boldsymbol{v}}_{2}&=\langle 5,2\rangle \cdot \langle 3,7\rangle \\ &=5\cdot 3+2\cdot 7 \\ &=15+14 \\ &=29 \end{align}[/latex]

To find the angle between them, we use the formula [latex]\cos \theta =\dfrac{\boldsymbol{v}}{|\boldsymbol{v}|}\cdot \dfrac{\boldsymbol{u}}{|\boldsymbol{u}|}[/latex].

[latex]\begin{align}\frac{\boldsymbol{v}}{|\boldsymbol{v}|}\cdot \frac{\boldsymbol{u}}{|\boldsymbol{u}|}&=\langle \frac{5}{\sqrt{29}}+\frac{2}{\sqrt{29}}\rangle \cdot \langle \frac{3}{\sqrt{58}}+\frac{7}{\sqrt{58}}\rangle \\ &=\frac{5}{\sqrt{29}}\cdot \frac{3}{\sqrt{58}}+\frac{2}{\sqrt{29}}\cdot \frac{7}{\sqrt{58}} \\ &=\frac{15}{\sqrt{1682}}+\frac{14}{\sqrt{1682}}=\frac{29}{\sqrt{1682}} \\ &=0.707107 \end{align}[/latex]

[latex]\theta={\cos }^{-1}\left(0.707107\right)=45^\circ[/latex]

Using the formula, [latex]\theta ={\cos }^{-1}\left(\dfrac{\boldsymbol{u}}{|\boldsymbol{u}|}\cdot \dfrac{\boldsymbol{v}}{|\boldsymbol{v}|}\right)[/latex],

[latex]\begin{align} \left(\frac{\boldsymbol{u}}{|\boldsymbol{u}|}\cdot \frac{\boldsymbol{v}}{|\boldsymbol{v}|}\right)&=\frac{-3\boldsymbol{i}+4\boldsymbol{j}}{5}\cdot \frac{5\boldsymbol{i}+12\boldsymbol{j}}{13} \\ &=\left(-\frac{3}{5}\cdot \frac{5}{13}\right)+\left(\frac{4}{5}\cdot \frac{12}{13}\right) \\ &=-\frac{15}{65}+\frac{48}{65} \\ &=\frac{33}{65}\end{align}[/latex]

[latex]\theta ={\cos }^{-1}\left(\frac{33}{65}\right) ={59.5}^{\circ }[/latex]