Let A and B be two distinct points with position vectors \(\vec{a}\) and \(\vec{b}\) respectively, with respect to the origin O. Which of the following statements are correct?
(A) \(\overrightarrow{AB} = \vec{b}-\vec{a}\)
(B) \(\overrightarrow{BA}\) is the negative of \(\overrightarrow{AB}\)
(C) \(|\overrightarrow{AB}| = |\overrightarrow{BA}|\)
(D) \(\overrightarrow{AB}+\overrightarrow{BA} = 2\,\overrightarrow{AB}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
By the triangle law in triangle OAB:
\(\overrightarrow{OA}+\overrightarrow{AB} = \overrightarrow{OB}\)
\(\overrightarrow{AB} = \vec{b}-\vec{a}\)
In the same way:
\(\overrightarrow{BA} = \vec{a}-\vec{b} = -\overrightarrow{AB}\)
So \(\overrightarrow{BA}\) is the negative vector of \(\overrightarrow{AB}\): same length, opposite direction, and \(\overrightarrow{AB}+\overrightarrow{BA} = \vec{0}\).
Checking the statements
(A) \(\overrightarrow{AB} = \vec{b}-\vec{a}\) — Correct. Position vector of the end point minus that of the starting point.
(B) \(\overrightarrow{BA}\) is the negative of \(\overrightarrow{AB}\) — Correct. \(\overrightarrow{BA} = \vec{a}-\vec{b} = -(\vec{b}-\vec{a})\).
(C) \(|\overrightarrow{AB}| = |\overrightarrow{BA}|\) — Correct. Both are the distance AB.
(D) \(\overrightarrow{AB}+\overrightarrow{BA} = 2\,\overrightarrow{AB}\) — Incorrect. The sum is \((\vec{b}-\vec{a})+(\vec{a}-\vec{b}) = \vec{0}\).