Vector Algebra
Statement-based problem on vector properties
nta_pyq_2023_jan
Grade 12
Question:
Let $\vec{a} = 2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}| = |\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot\vec{c}=0$. Consider the following two statements: (A) $|\vec{a}+\lambda\vec{c}| \geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. (B) $\vec{a}$ and $\vec{c}$ are always parallel. Then:
only (B) is correct
neither (A) nor (B) is correct
only (A) is correct
both (A) and (B) are correct
Step-by-Step Solution
Key Concept: The equal-magnitude condition gives $\vec{a}\cdot\vec{c}=0$. Use this to evaluate statements A and B.
$\vec{a}\cdot\vec{c}=0$. (A) is true (since $\lambda^2|\vec{c}|^2 \geq 0$). (B) is false ($\vec{a}\perp\vec{c}$, not parallel). Answer: (3)
Correct Answer: only (A) is correct