For any vector a = a 1 i + a 2 j + a 3 k , with 10 | a i |1 , i = 1 , 2 , 3 , consider the following statements: ( A ) : max { | a 1 | , | a 2 | , | a 3 | } | a| ( B ) : | a | 3 ? max { | a 1 | , | a 2 | , | a 3 | }

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Solution

Q.
For any vector $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k},$ with $10$ $| a_{i} |$ $< 1, i = 1, 2, 3,$ consider the following statements:

$(A)$ : $\max {| a_{1} |, | a_{2} |, | a_{3} |} \leq | \vec{a} |$

$(B)$ : $| \vec{a} |$ $\leq 3 \max {| a_{1} |, | a_{2} |, | a_{3} |}$ [2023]

1 Only (B) is true

2 Only (A) is true

3 Neither (A) nor (B) is true

4 Both (A) and (B) are true

Ans.
(4)

$Without loss of generality, let | a_{1} | \leq | a_{2} | \leq | a_{3} |$

$| \vec{a} |^{2} = | a_{1} |^{2} + | a_{2} |^{2} + | a_{3} |^{2} \geq | a_{3} |^{2}$

$\Rightarrow | \vec{a} | \geq | a_{3} | = \max {| a_{1} |, | a_{2} |, | a_{3} |}$

$(A) is true$

$| \vec{a} |^{2} = | a_{1} |^{2} + | a_{2} |^{2} + | a_{3} |^{2} \leq | a_{3} |^{2} + | a_{3} |^{2} + | a_{3} |^{2}$

$\Rightarrow | \vec{a} |^{2} \leq 3 | a_{3} |^{2}$

$\Rightarrow | \vec{a} | \leq \sqrt{3} | a_{3} | = \sqrt{3} \max {| a_{1} |, | a_{2} |, | a_{3} |} \leq 3 \max {| a_{1} |, | a_{2} |, | a_{3} |}$

$(B) is true$

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