\(\rm \vec u=\ \textless \ 7,2\ \textgreater \ \qquad\qquad\qquad v=\ \textless \ 0,-4\ \textgreater \\)
Recall that the cosine of the angle between the vectors is given by,
\(\rm \cos\theta=\dfrac{\vec u\cdot\vec v}{|u||v|}\)
So we have a bunch of things we need to do.
Find the dot product of u and v,
\(\rm \vec u\cdot \vec v=\ \textless \ 7,2\ \textgreater \ \cdot\ \textless \ 0,-4\ \textgreater \ =7(0)+2(-4)=-8\)
That gives us our numerator,
\(\rm \cos\theta=\dfrac{-8}{|u||v|}\)
Find the magnitude of each vector,
\(\rm |u|=\sqrt{7^2+2^2}=\sqrt{53}\qquad\qquad |v|=\sqrt{0^2+(-4)^2}=4\)
Ok that gives us our denominator,
\(\rm \cos\theta=\dfrac{-8}{4\sqrt{53}}\)
To find your angle theta, apply inverse cosine,
\(\rm \cos^{-1}\left(\dfrac{-8}{4\sqrt{53}}\right)=\theta\)
Let your calculator do the rest.
Hope that helps!
