question_answer

If \[\vec{a},\] \[\vec{b}\] and \[\vec{c}\] are three vectors, such that \[|\vec{a}|\,\,=5,\] \[|\vec{b}|\,\,=12,\] \[|\vec{c}|\,\,=13\]and \[\vec{a}+\vec{b}+\vec{c}=\vec{0}.\] find the value of \[\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}\].

Answer:

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