The question is as follows: 
I started to approach this by thinking of the vectors $v$ and $w$ as of the form $a \hat i+b \hat j$ and $c \hat i+d \hat j$. Based on this, I created three equations:
$\sqrt{a^2+b^2} = 3$
and
$\sqrt{c^2+d^2} = \sqrt{2}$ from the vector magnitudes that are given, and $ac+bd=3$ from the vector angle equation $\hat u \cdot \hat v = |\hat u||\hat v|cos( \theta)$.
However, as there are four variables, and only three equations, I am obviously missing something here in order to solve the problem. What vector equation am I missing to be able to determine the vectors, and therefore this arbitrary magnitude which the question asks for?
Best Answer
You have\begin{align}|\mathbf v+2\mathbf w|^2&=|\mathbf v|^2+4|\mathbf w|^2+4\langle\mathbf v,\mathbf w\rangle\\&=17+4\langle\mathbf v,\mathbf w\rangle.\end{align}Furthermore, since the angle between $\mathbf v$ and $\mathbf w$ is $\frac\pi4$,$$\langle\mathbf v,\mathbf w\rangle=|\mathbf v||\mathbf w|\cos\left(\frac\pi4\right)=3.$$Therefore, $|\mathbf v+2\mathbf w|^2=29$.