Q.In triangle $ABC$ (Fig 10.18), 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) $\vec{AB} - \vec{CB} + \vec{CA} = \vec{0}$

Concept: Vector Addition Triangle Law — in a closed triangle, the head-to-tail sum of the directed sides taken in order is zero.

Reasoning:

1. For triangle $ABC$, the natural cycle $\vec{AB} + \vec{BC} + \vec{CA}$ returns to the starting point, so the sum is $\vec{0}$. Hence (A) is true.
2. Since $\vec{CA} = -\vec{AC}$, statement (B) becomes $\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}$, which is the same as (A). So (B) is true. …

The core idea here is the Triangle Law of Vector Addition: if you place vectors head to tail, the resultant vector goes from the first tail to the last head. When you go completely around a closed polygon (like a triangle) and return to the starting point, the net displacement is zero. That is, for any triangle $ABC$,

This is the fundamental relation. Every option is just a rearrangement of these three vectors. Our job is to check which one does not reduce to $\vec{0}$.

Let’s go through each option one by one.

1. Option (A): $\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}$

   This is exactly the closed-triangle condition. It is true by definition.

2. Option (B): $\vec{AB} + \vec{BC} - \vec{AC} = \vec{0}$

   Notice that $-\vec{AC} = \vec{CA}$ (reversing a vector flips its direction). So the expression becomes $\vec{AB} + \vec{BC} + \vec{CA}$, which is exactly option (A). Hence it is true.

3. Option (C): $\vec{AB} + \vec{BC} - \vec{CA} = \vec{0}$

   Here $-\vec{CA} = \vec{AC}$. So the expression is $\vec{AB} + \vec{BC} + \vec{AC}$.

   But $\vec{AB} + \vec{BC} = \vec{AC}$ (by the triangle law, going from A to B to C gives the same as going directly from A to C). So this sum becomes $\vec{AC} + \vec{AC} = 2\vec{AC}$, which is not zero unless the triangle is degenerate. Therefore this statement is false. …

Method: Deciding Which Triangle-Law Identity is False

This is a reasoning task on the triangle law of vector addition. The technique is to reduce every option to the closed-loop form and see which one fails to become $\vec{0}$.

Steps

Step 1: State the anchor identity.

Going around a closed triangle returns you to the start:

Step 2: Rewrite every reversed vector using $-\vec{PQ}=\vec{QP}$.

Whenever you see a minus sign in front of a directed vector, flip its letters: $-\vec{AC}=\vec{CA}$, $-\vec{CB}=\vec{BC}$, and so on. This exposes hidden closed loops.

Step 3: Compare each option to the anchor. …

Common Mistakes

Mistake 1: Treating $-\vec{CA}$ as $\vec{CA}$.

Why it's wrong: reversing a vector flips its letters, so $-\vec{CA}=\vec{AC}$. Forgetting this turns the false option (C) into the true identity by accident. Correct approach: apply $-\vec{PQ}=\vec{QP}$ before simplifying.

Mistake 2: Assuming $\vec{AB}+\vec{BC}=\vec{0}$. …