Question

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

Answer 1

Let \[\text { ABC }\] be a triangle such that \[\overrightarrow {BC} = \vec{a}\] \[\overrightarrow{AB} = \vec{c}\] and \[\overrightarrow{CA} = \vec{b}\]. Then,    \[\vec{a} + \vec{b} + \vec{c} = \overrightarrow{BC} + \overrightarrow{CA} +\overrightarrow{AB}\] 

\[\vec{a} + \vec{b} + \vec{c} = \overrightarrow{BA} +  \overrightarrow{AB}\]                     

 [∵ \[\overrightarrow{BC} + \overrightarrow{CA} = \overrightarrow{BA}\]] 
\[\Rightarrow \vec{a} + \vec{b} + \vec{c} = \overrightarrow{BB}\]                               [ Using triangle law]
\[\Rightarrow \vec{a} + \vec{b} + \vec{c}\to = \vec{0}\]                                  [ By definition of null vector]
Other possibilities are
\[\left( i \right) \vec{c} + \vec{a} = \vec{b}\]
\[\left( ii \right) \vec{a} + \vec{b} = \vec{c}\]
\[\left( iii \right) \vec{b} + \vec{c} = \vec{a}\]