vector $a=mi + j$ and vector $b=i + mj$. Find $m$ given the acute angle between them is $30^{\circ}$

Question

I've tried using the formula $mi+j = |a|\cos 30^{\circ} + |b|\sin 30^{\circ}$ and adding them together to get $(mi+j) + (i+mj)$.

I'm pretty sure you have to turn it into a quadratic to get 2 answers, but I can't get it into quadratic formula.

Answer 1

Note,

$$\cos30=\frac{\sqrt3}2=\frac {a\cdot b}{|a||b|}=\frac{2m}{1+m^2}$$ Solve to get,

$$m=\frac 1{\sqrt3},\>\>\>\>\>m=\sqrt3$$