$open vertical bar table row 1 cell space space space straight a space end cell cell space bc end cell row 1 cell space space straight b end cell cell space space ca end cell row 1 cell space space straight c end cell cell space space ab end cell end table close vertical bar space equals space open vertical bar table row 1 cell space space space straight a space end cell cell space space space straight a squared end cell row 1 cell space space straight b end cell cell space space straight b squared end cell row 1 cell space space straight c end cell cell space space straight c squared end cell end table close vertical bar space$ Without expanding, prove that

Solution

Let space space increment space equals space space open vertical bar table row 1 cell space space space straight a space end cell cell space bc end cell row 1 cell space space straight b end cell cell space space ca end cell row 1 cell space space straight c end cell cell space space ab end cell end table close vertical bar space equals space begin inline style 1 over abc end style space space open vertical bar table row cell space space space straight a space end cell cell space space space straight a squared space end cell cell space space abc end cell row cell space space straight b end cell cell space space straight b squared end cell cell space space abc end cell row cell space space straight c end cell cell space space straight c squared end cell cell space space abc end cell end table close vertical bar space by space multiplying space straight R subscript 1 comma space end subscript straight R subscript 2 comma straight R subscript 3 space by space straight a comma space straight b comma straight c space respectively space space space space space space space space space space space space space equals space begin inline style abc over abc end style space open vertical bar table row straight a cell space space space space space straight a squared end cell cell space space space bc end cell row straight b cell space space space space straight b squared end cell cell space space space ca end cell row straight c cell space space space straight c squared end cell cell space space space space ab end cell end table close vertical bar space equals space begin inline style 1 over abc end style space space open vertical bar table row 1 cell space space space straight a space end cell cell space space space straight a squared space end cell row 1 cell space space straight b end cell cell space space straight b squared end cell row 1 cell space space straight c end cell cell space space straight c squared end cell end table close vertical bar

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