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प्रश्न
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?
उत्तर
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
\[\left( i \right) \vec{c} + \vec{a} = \vec{b}\]
\[\left( ii \right) \vec{a} + \vec{b} = \vec{c}\]
\[\left( iii \right) \vec{b} + \vec{c} = \vec{a}\]
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