Given the direction cosines of two mutually perpendicular lines, show the direction cosines of the lines perpendicular to the above two lines are:

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This is the solution given in book

I just want to ask that if it is written two equations of relations of l, l1, m, etc. How did he write it in that ratio or fractional form like l upon something and m upon something etc and made it equal. How did he solve that step? How is it written directly? If there is some step then let me know. I know the way to prove it by vector method. But in that solution particularly how did he write it directly, is there a silly step that I dont know?