In order to determine the angle between the given vectors, use the following formula:
[tex]\vec{u}\cdot\vec{v}=uv\cos \theta[/tex]where the left hand side of the equation is the dot product of u and v vectors, u and v are the magnitude of the vectors and θ is the angle between the vectors.
Then, by solving for θ, you obtain:
[tex]\begin{gathered} \cos \theta=\frac{\vec{u}\cdot\vec{v}}{uv} \\ \theta=\cos ^{-1}(\frac{\vec{u}\cdot\vec{v}}{uv}) \end{gathered}[/tex]Then, first calculate the dot product:
[tex]\begin{gathered} \vec{u}\cdot\vec{v}=(u_x)(v_x)+(u_y)(v_y) \\ \vec{u}\cdot\vec{v}=(0)(11)+(-10)(-12)=120 \end{gathered}[/tex]The magnitudes of the vectors are:
[tex]\begin{gathered} u=\sqrt[]{u^2_x+u^2_y}=\sqrt[]{(0)^{}+(-10)^2}=\sqrt[]{100}=10.00 \\ v=\sqrt[]{v^2_x+v^2_y}=\sqrt[]{(11)^2+(-12)^2}=\sqrt[]{265}\approx16.28 \end{gathered}[/tex]Then, by replacing the dot product and the valued of u and v into the expression for θ, you obtain:
[tex]\theta=\cos ^{-1}(\frac{120}{10.00\cdot16.28})\approx42.51[/tex]Hence, the angle between u and v vectors is approximately 42.51 degrees.
