In ΔABC: (A) AB+BC+CA=O (B) AB+BC-AC=O (C) AB+BC-CA=O (D) AB-CB+CA=O…

The correct answer is Option (1) → (A), (B), (D) only

By the triangle law of vector addition:

    \(\vec{AB}+\vec{BC} = \vec{AC}\)

Reversing a vector changes its sign:

    \(\vec{CA} = -\vec{AC}\)
    \(\vec{CB} = -\vec{BC}\)

So \(\vec{AB}+\vec{BC} = \vec{AC}\), with \(\vec{CA} = -\vec{AC}\) and \(-\vec{CB} = \vec{BC}\).

Checking the statements

(A) \(\vec{AB}+\vec{BC}+\vec{CA}=\vec{O}\) — Correct. The left side is \(\vec{AC}+\vec{CA} = \vec{O}\).

(B) \(\vec{AB}+\vec{BC}-\vec{AC}=\vec{O}\) — Correct. The left side is \(\vec{AC}-\vec{AC} = \vec{O}\).

(C) \(\vec{AB}+\vec{BC}-\vec{CA}=\vec{O}\) — Incorrect. The left side is \(\vec{AC}+\vec{AC} = 2\vec{AC}\), not \(\vec{O}\).

(D) \(\vec{AB}-\vec{CB}+\vec{CA}=\vec{O}\) — Correct. \(-\vec{CB} = \vec{BC}\), so the left side is \(\vec{AB}+\vec{BC}+\vec{CA} = \vec{AC}+\vec{CA} = \vec{O}\).

(E) \(\vec{AB}-\vec{CB}-\vec{CA}=\vec{O}\) — Incorrect. The left side is \(\vec{AB}+\vec{BC}+\vec{AC} = 2\vec{AC}\), not \(\vec{O}\).