Question

Give a condition that three vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ form the three sides of a triangle. What are the other possibilities?

Answer 1

Let $ABC$ be a triangle such that $\overrightarrow{BC}=\overrightarrow{a}$, $\overrightarrow{AB}=\overrightarrow{c}$ and $\overrightarrow{CA}=\overrightarrow{b}$. Then,
$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{BC}+\overrightarrow{CA}+\overrightarrow{AB}$
$\Rightarrow \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{BA}+\overrightarrow{AB}$ [∵ $\overrightarrow{BC}+\overrightarrow{CA}=\overrightarrow{BA}$]
$\Rightarrow \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{BB}$ [ Using triangle law]
$\Rightarrow \overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ [ By definition of null vector]

Other possibilities are
$\left(\mathrm{i}\right)\overrightarrow{c}+\overrightarrow{a}=\overrightarrow{b}$
$\left(\mathrm{ii}\right)\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{c}$
$\left(\mathrm{iii}\right)\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{a}$