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If $a, b, c$ are three non-zero vectors such that $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=0$ and $m=\boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b} \cdot \boldsymbol{c}+\boldsymbol{c} \cdot \boldsymbol{a}$, then
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Verified Answer
The correct answer is:
$m < 0$
$$
\begin{array}{l}
\text { Here, } \mathbf{a}+\mathbf{b}+\mathbf{c}=0 \\
\Rightarrow|\mathbf{a}+\mathbf{b}+\mathbf{c}|^{2}=0 \\
\Rightarrow|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+\mid \mathbf{c}^{2}+2\{\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}\}=0 \\
\Rightarrow \mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a} \\
=-\frac{1}{2}\left\{|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+|\mathbf{c}|^{2}\right\} < 0 \\
\Rightarrow \mathrm{m} < 0
\end{array}
$$
\begin{array}{l}
\text { Here, } \mathbf{a}+\mathbf{b}+\mathbf{c}=0 \\
\Rightarrow|\mathbf{a}+\mathbf{b}+\mathbf{c}|^{2}=0 \\
\Rightarrow|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+\mid \mathbf{c}^{2}+2\{\mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a}\}=0 \\
\Rightarrow \mathbf{a} \cdot \mathbf{b}+\mathbf{b} \cdot \mathbf{c}+\mathbf{c} \cdot \mathbf{a} \\
=-\frac{1}{2}\left\{|\mathbf{a}|^{2}+|\mathbf{b}|^{2}+|\mathbf{c}|^{2}\right\} < 0 \\
\Rightarrow \mathrm{m} < 0
\end{array}
$$
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