Question

In triangle ABC which of the following is not true.

• A. \(\vec{AB}+\vec{BC}+\vec{CA}=\vec{0}\)
• B. \(\vec{AB}+\vec{BC}-\vec{AC}=\vec{0}\)
• C. \(\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}\)
• D. \(\vec{AB}-\vec{CB}+\vec{CA}=\vec{0}\)

Answer 1

The Correct Option is C

The correct answer is C:\(\vec{AB}+\vec{BC}-\vec{CA}=\vec{0}\)

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