Vector Algebra
Vector Addition
Grade None
Question:
<p>Consider points \(A,B,C\) with position vectors \(\vec{a},\vec{b},\vec{c}\) respectively.</p>
<p><strong>Statement-1:</strong> \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}=\vec{0}\).</p>
<p><strong>Statement-2:</strong> \(A,B,C\) form the vertices of a triangle.</p>
<li>S-1 True, S-2 True; S-2 IS a correct explanation for S-1</li>
<li>S-1 True, S-2 True; S-2 is NOT a correct explanation for S-1</li>
<li>S-1 True, S-2 False</li>
<li>S-1 False, S-2 True</li>
Step-by-Step Solution
Key Concept: Statement-1 is trivially true for ANY three points (it's a vector identity). Statement-2 is false because A, B, C could be collinear.
Statement-1: \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA} = (\vec{b}-\vec{a})+(\vec{c}-\vec{b})+(\vec{a}-\vec{c})=\vec{0}\). Always true for any 3 points. ✓
Statement-2: A, B, C may be collinear (e.g., A=(0,0,0), B=(1,0,0), C=(2,0,0)). They don't necessarily form a triangle. ✗
So Statement-1 is True, Statement-2 is False. Answer: (C)
Correct Answer: C