If \(\rm \vec a.\vec b=0\) and \(\vec a+\vec b\) makes an angle of 30° with \(\vec a\) then

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AWES TGT 2015: Mathematics
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  1. \(|\vec b|=\sqrt3 |\vec a|\)
  2. \(|\vec a|=\sqrt3 |\vec b|\)
  3. \(|\vec a|=2 |\vec b|\)
  4. \(|\vec b|=2 |\vec a|\)

Answer (Detailed Solution Below)

Option 2 : \(|\vec a|=\sqrt3 |\vec b|\)
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AWES PGT 2012 - History Official Paper
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Detailed Solution

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Concept:

  • Two vectors a and b are perpendicular to each other if a . b = o

Calculations:

Given \(\rm \vec a.\vec b=0\) and \(\vec a+\vec b\) makes an angle of 30° with \(\vec a\) 

We can understand it better by the below image.

F1 AbhishekP Madhuri 24.01.2022 D4

We have a . b = o, i.e., vector a is perpendicular to vector b

Clearly, from the above diagram Vectors a, b, and a + b together form a right-angled triangle 

So, in \(\bigtriangleup PQR\), we have

\(tan \ 30^{o}=\frac{|b|}{|a|}\)

⇒ \(\frac{1}{√{3}}=\frac{|b|}{|a|}\)

⇒ |a| = √3 |b|

Hence, the correct answer is option 2).

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