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Vector Algebra Part 2

Section: Mathematics

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Question : 16 of 100

Marks: +1, -0

For any vector

\overrightarrow{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}

10|a_i|<1,\ i=1,2,3

, consider the following statements :

\max\{|a_1|,|a_2|,|a_3|\}\leq|\overrightarrow{a}|

|\overrightarrow{a}|\leq 3\max\{|a_1|,|a_2|,|a_3|\}

[11-Apr-2023 shift 1]

Only (B) is true
Both (A) and (B) are true
Neither (A) nor (B) is true
Only (A) is true

Solution:

Without loss of generality

Let

|a_1|\leq|a_2|\leq|a_3|

\;|\overrightarrow{a}|^2=|\overrightarrow{a}_1|^2+|\overrightarrow{a}_2|^2+|\overrightarrow{a}_3|^2\geq (a_3)^2

\;\Rightarrow|\overrightarrow{a}|\geq|a_3|=\max\{|a_1|,|a_2|,|a_3|\}

A

is true

\;|\overrightarrow{a}|^2=|a_1|^2+|a_2|^2+|a_3|^2\leq|a_3|^2+|a_3|^2+|a_3|^2

\;\Rightarrow|\overrightarrow{a}|^2\leq 3|a_3|^2

\;\Rightarrow|\overrightarrow{a}|\leq \sqrt{3}|\overrightarrow{a}_3|=\sqrt{3}\max\{|a_1|,|a_2|,|a_3|\}

\leq 3\max\{|a_1|,|a_2|,|a_3|\}

(2) is true

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