Question

If \(m_1,m_2,m_3,m_4\) are respectively the magnitudes of the vectors
\[ \vec{a_1}=2\hat{i}-\hat{j}+\hat{k},\quad \vec{a_2}=3\hat{i}-4\hat{j}-4\hat{k}, \] \[ \vec{a_3}=\hat{i}+\hat{j}-\hat{k},\quad \vec{a_4}=-\hat{i}+3\hat{j}+\hat{k} \] then

• A. \(m_3
• B. \(m_1
• C. \(m_3
• D. \(m_3

Hint:

Magnitude of \((a,b,c)\) is \(\sqrt{a^2+b^2+c^2}\). Compare using squared values to avoid approximation errors.

Answer 1

The Correct Option is A

Step 1: Compute each magnitude.
\[ m_1=\sqrt{2^2+(-1)^2+1^2}=\sqrt{4+1+1}=\sqrt{6} \] 
\[ m_2=\sqrt{3^2+(-4)^2+(-4)^2}=\sqrt{9+16+16}=\sqrt{41} \] 
\[ m_3=\sqrt{1^2+1^2+(-1)^2}=\sqrt{3} \] 
\[ m_4=\sqrt{(-1)^2+3^2+1^2}=\sqrt{1+9+1}=\sqrt{11} \] 
Step 2: Arrange in increasing order. 
\[ \sqrt{3}<\sqrt{6}<\sqrt{11}<\sqrt{41} \] 
So: 
\[ m_3 Final Answer: 
 

\[\boxed{m_3}\]