Answer:
We have,\[|\overrightarrow{a}|\,\,=5,\]\[|\overrightarrow{b}|\,\,=12\]and \[|\overrightarrow{c}|\,\,=13\] Also, \[\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0\] Now, \[\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{b}\cdot \overrightarrow{c}+\overrightarrow{c}\cdot \overrightarrow{a}\] \[=\frac{1}{2}(|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}{{|}^{2}}-|\overrightarrow{a}{{|}^{2}}-|\overrightarrow{b}{{|}^{2}}-|\overrightarrow{c}{{|}^{2}})\] \[=-\frac{1}{2}[{{(5)}^{2}}+{{(12)}^{2}}+{{(13)}^{2}}]\] \[=-\frac{1}{2}(25+144+169)\] \[=-\frac{1}{2}(2\times 169)=-169\]
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