Q:

Vector dot product - determine the angle

Im having problem determining cosines. Does anyone know how to solve it? Can anyone provide some hint or full solution? 
Please Help !

Data are the vectors $x$ and $y$, where $\| x \| = 3$, $\|y\| = 4$, and $xy =
 2$. Determine $\| x - 2y \|$, $\| x + 3y \|$ and $cos (∡ (x - 2y, 3y + x))$.

A:

For example, and denoting by $\;\langle x,y\rangle\;$ the dot product of $\;x,y\;$:
$$\left\|x-2y\right\|^2=\langle x-2y,\,x-2y\rangle=\langle x,x\rangle-4\langle x,y\rangle+4\langle y,y\rangle=9-4\cdot2+4\cdot16=65$$
Do the rest, and also remember that if $\;\theta=\angle(a,b)\;$ ,then
$$\cos\theta=\frac{\langle a,b\rangle}{\left\|a\right\|\left\|b\right\|}$$

