Question

In Figure, which of the following is not true?

Options

• \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} = \vec{0}\]

• \[\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AC} = \vec{0}\]

• \[\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{CA} = \vec{0}\]

• \[\overrightarrow{AB} - \overrightarrow{CB} + \overrightarrow{CA} = \vec{0}\]

Answer 1

\[\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{CA} = \vec{0}\]

We have, LHS =  \[\overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{CA} = \overrightarrow{AC} - \overrightarrow{CA}\] [∵  \[\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\] 

\[= - \overrightarrow{CA} - \overrightarrow{CA} \]

\[ = - 2 \overrightarrow{CA}\]

So, LHS \[\neq\] RHS
Hence, It is not true.