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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
If $a, b, c$  are mutually perpendicular vectors of equal magnitudes, then the angle between the vectors $a$ and $a + b + c$ is
  • A
    $\frac{\pi }{3}$
  • B
    $\frac{\pi }{6}$
  • ✓
    ${\cos ^{ - 1}}\frac{1}{{\sqrt 3 }}$
  • D
    $\frac{\pi }{2}$
✓

Answer

Correct option: C.
${\cos ^{ - 1}}\frac{1}{{\sqrt 3 }}$
c
(c) Since $a,\,\,b$ and $c$ are mutually perpendicular, so $a\,.\,b = b\,.\,c = c\,.\,a = 0$

Angle between $a$ and $a + b + c$ is $\cos \theta = \frac{{a.(a + b + c)}}{{|a||a + b + c|}}$ .....$(i)$

Now $|a| = |b| = |c| = a$

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

$ = {a^2} + {a^2} + {a^2} + 0 + 0 + 0$

$\Rightarrow  |a + b + c{|^2} = 3{a^2} \Rightarrow |a + b + c| = \sqrt 3 a$

Putting this value in $(i),$ we get $\theta = {\cos ^{ - 1}}\frac{1}{{\sqrt 3 }}.$

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