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=(\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=(-8)/(|u||v|)`

Find the magnitude of each vector,

`\rm |u|=√(7^2+2^2)=√(53)\qquad\qquad |v|=√(0^2+(-4)^2)=4`

Ok that gives us our denominator,

`\rm \cos\theta=(-8)/(4√(53))`

To find your angle theta, apply inverse cosine,

`\rm \cos^(-1)\left((-8)/(4√(53))\right)=\theta`

Let your calculator do the rest.
Hope that helps!