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Mathematics Question on Vector Algebra

In triangle ABC which of the following is not true.
Vector

A

AB⃗+BC⃗+CA⃗=0⃗\vec{AB}+\vec{BC}+\vec{CA}=\vec{0}

B

AB⃗+BC⃗−AC⃗=0⃗\vec{AB}+\vec{BC}-\vec{AC}=\vec{0}

C

AB⃗+BC⃗−CA⃗=0⃗\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}

D

AB⃗−CB⃗+CA⃗=0⃗\vec{AB}-\vec{CB}+\vec{CA}=\vec{0}

Answer

AB⃗+BC⃗−CA⃗=0⃗\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}

Explanation

Solution

The correct answer is C:AB⃗+BC⃗−CA⃗=0⃗\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}
Vector
On applying the triangle law of addition in the given triangle,we have:
AB⃗+BC⃗=AC⃗\vec{AB}+\vec{BC}=\vec{AC}
AB⃗+BC⃗=−CA⃗\vec{AB}+\vec{BC}=\vec{-CA}
AB⃗+BC⃗+CA⃗=0⃗\vec{AB}+\vec{BC}+\vec{CA}=\vec{0}
∴The equation given in alternative A is true.
AB⃗+BC⃗=AC⃗\vec{AB}+\vec{BC}=\vec{AC}
AB⃗+BC⃗−AC⃗=0⃗\vec{AB}+\vec{BC}-\vec{AC}=\vec{0}
∴The equation given in alternative B is true.
From equation(2),we have:
AB⃗−CB⃗+CA⃗=0⃗\vec{AB}-\vec{CB}+\vec{CA}=\vec{0}
∴The equation given in alternative D is true.
Now,consider the equation given in alternative C:
AB⃗+BC⃗−CA⃗=0⃗\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}
AB⃗+BC⃗=CA⃗\vec{AB}+\vec{BC}=\vec{CA}
From equation(1)and (3),we have:
AC⃗=CA⃗\vec{AC}=\vec{CA}
⇒AC⃗=−AC⃗⇒\vec{AC}=\vec{-AC}
⇒AC⃗+AC⃗=0⃗⇒\vec{AC}+\vec{AC}=\vec{0}
⇒2AC⃗=0⃗⇒2\vec{AC}=\vec{0}
⇒AC⃗=0⃗⇒\vec{AC}=\vec{0},Which is not true.
Hence,the equation given in alternative C is incorrect.
The correct answer is C.