In this cutout of the book the author is showing about voltage source inverter modulation technique. He is basically transforming the axis to a newer representation.
I cannot understand how to get the shown expressions for m and n
The total angle is 60°.
Just write $\vec i$ and $\vec j$ in terms of $\hat x$ and $\hat y$. $$\vec i=a\hat x\\\vec j=a\frac12 \hat x+a\frac{\sqrt 3}2\hat y$$ Then you know $$V=\alpha \hat x+\beta\hat y=m\vec i+n\vec j=a(m+\frac n2)\hat x+a\frac{\sqrt 3}2n\hat y$$ You get $$\alpha=a(m+n/2)\\\beta=an\sqrt 3/2$$ Find $m,n$