In order to calculate the angle between the vectors, we need to use the dot product:
[tex]u\cdot v=|u||v|\cos\theta=u_xv_x+u_yv_y[/tex]
Calculating the magnitude of each vector, we have:
[tex]\begin{gathered} |u|=\sqrt{u_x^2+u_y^2}=\sqrt{49+4}=\sqrt{53}\\ \\ |v|=\sqrt{v_x^2+v_y^2}=\sqrt{9+16}=\sqrt{25}=5 \end{gathered}[/tex]
Now, calculating the dot product, we have:
[tex]u\cdot v=u_xv_x+u_yv_y=(-7)\cdot(-3)+2\cdot(-4)=21-8=13[/tex]
So, calculating the cosine of theta and then the angle theta, we have:
[tex]\begin{gathered} \cos\theta=\frac{u\cdot v}{|u||v|}=\frac{13}{5\sqrt{53}}=\frac{13}{36.4}=0.35714\\ \\ \theta=\cos^{-1}(0.35714)\\ \\ \theta=69.07° \end{gathered}[/tex]
