In Figure, which of the following is not true?

(a) $\vec{A B} + \vec{B C} + \vec{C A} = \vec{0}$

(b) $\vec{A B} + \vec{B C} - \vec{A C} = \vec{0}$

(c) $\vec{A B} + \vec{B C} - \vec{C A} = \vec{0}$

(d) $\vec{A B} - \vec{C B} + \vec{C A} = \vec{0}$
Figure

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Solution

(c) $\vec{A B} + \vec{B C} - \vec{C A} = \vec{0}$

We have, LHS = $\vec{A B} + \vec{B C} - \vec{C A} = \vec{A C} - \vec{C A}$ [∵ $\vec{A B} + \vec{B C} = \vec{A C}$ ]
$= - \vec{C A} - \vec{C A} = - 2 \vec{C A}$
So, LHS $\neq$ RHS
Hence, It is not true.

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