[tex]a = 2\sqrt 3 i+2mj\\ \\ b = mi+\sqrt 3 j\\ \\ \cos(45) =\dfrac{a\cdot b}{|a|\cdot |b|} \\ \\ \\\dfrac{\sqrt 2}{2} = \dfrac{2\sqrt 3 m+2\sqrt 3 m}{\sqrt{12+4m^2}\cdot \sqrt{m^2+3}} \\ \\\dfrac{\sqrt 2}{2} = \dfrac{4\sqrt 3 m}{2\sqrt{m^2+3}\cdot \sqrt{m^2+3}} \\ \\\dfrac{\sqrt 2}{2} = \dfrac{2\sqrt 3m}{m^2+3}[/tex]
[tex](m^2+3)\sqrt 2 = 4\sqrt 3 m \\ \\ 2(m^2+3) = 4\sqrt 6 m\\ \\ m^2-2\sqrt 6m+3 = 0 \\ \\ (m-\sqrt 6)^2- 3= 0 \\ \\ (m-\sqrt 6 - \sqrt 3)(m-\sqrt 6 +\sqrt 3) = 0 \\ \\ \Rightarrow \boxed{m = \sqrt 6 \pm \sqrt 3}[/tex]
