Question

If $\overline{a},\overline{b},\overline{c}$ are three non-zero vectors such that $\overline{a}+\overline{b}+\overline{c}=0$ then the value of $\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}$ is

• A. less than zero
• B. equal to zero
• C. greater than zero
• D. 3

Answer 1

The correct option is A less than zero

Given $\overline{a},\overline{b},\overline{c}$ are three non-zero vectors such that $\overline{a}+\overline{b}+\overline{c}=0$
$\Rightarrow (\overline{a}+\overline{b}+\overline{c})\cdot (\overline{a}+\overline{b}+\overline{c})=0$
$\Rightarrow |\overline{a}{|}^2+|\overline{b}{|}^2+|\overline{c}{|}^2+2(\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a})=0$
$\Rightarrow \overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}=-\frac{1}{2}(|\overline{a}{|}^2+|\overline{b}{|}^2+|\overline{c}{|}^2)$
$\therefore \overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}<0$
Hence, option A.