Vector Algebra
Norm Inequalities for Vectors
nta_pyq_2023_apr
Grade 12
Question:
For $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ with $10|a_i|<1$: (A) $\max|a_i|\leq|\vec{a}|$ and (B) $|\vec{a}|\leq3\max|a_i|$
Only (B) is true
Only (A) is true
Both (A) and (B) are true
Neither (A) nor (B) is true
Step-by-Step Solution
Key Concept: (A): $\max|a_i|\leq\sqrt{a_1^2+a_2^2+a_3^2}=|\vec{a}|$ — always true. (B): $|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}\leq\sqrt{3}\max|a_i|\leq3\max|a_i|$ — true.
Both true.
Correct Answer: 3