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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
The vector $a + b$ bisects the angle between the vectors $ a $ and $ b,$  if
  • A
    $|a|\, = \,|b|$
  • ✓
    $|a|\, = \,|b|$ or angle between a and b is zero
  • C
    $|a|\,\, = m\,|b|$
  • D
    None of these
✓

Answer

Correct option: B.
$|a|\, = \,|b|$ or angle between a and b is zero
b
(b) Since the angle between $a + b$ and $a$ and the angle between $a + b$ and $b$  are the same, so we have

$\frac{{(a + b)\,.\,a}}{{|a + b|\,|a|}} = \frac{{(a + b)\,.\,b}}{{|a + b|\,|b|}}$

$ \Rightarrow \frac{{|a{|^2}}}{{|a + b|\,|a|}} + \frac{{b\,.\,a}}{{|a + b|\,|a|}} = \frac{{a\,.\,b}}{{|a + b|\,|b|}} + \frac{{|b{|^2}}}{{|a + b|\,|b|}}$

$ \Rightarrow \frac{{|a| - |b|}}{{|a + b|}}\left( {1 - \frac{{a\,.\,b}}{{|a|\,|b|}}} \right) = 0$

Hence $|a|\, = \,|b|$ or angle between $a$ and $b$ is $0$.

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