Question

Find the angle between the vectors u = 7i +2j and v = -4j

Answer 1

$\rm \vec u=\ \textless \ 7,2\ \textgreater \ \qquad\qquad\qquad v=\ \textless \ 0,-4\ \textgreater$

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!