$ \Rightarrow \,|\,\vec a + \;\vec b + \,\vec c {|^2}\,\, = \,\,0$
$ \Rightarrow \,\,\,|\,\vec a |{\,^2}\,\, + \;\,|\,\vec b {|^2}\, + \,\,|\,\vec c {|^2}\, + \,\,2\,\,\left( {\vec a .\,\vec b + \,\vec b .\,\vec c \, + \;\vec c .\,\,\vec a } \right)\,\, = \,\,0$
$ \Rightarrow \,\,1\,\, + \;\,4\,\, + \;\,9\,\, + \;\,2\,\left( {\vec a .\,\vec b + \,\vec b .\,\vec c \, + \;\vec c .\,\,\vec a } \right)\,\, = \,\,0\,\,$
$ \Rightarrow \,\,\vec a .\,\vec b + \,\vec b .\,\vec c \, + \;\vec c .\,\,\vec a \,\, = \,\, - 7$
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