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10. vector algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths10. vector algebra1 Mark
MCQ
Let $\vec{a}, \vec{b}, \vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36 \cos ^{2} 2 \theta$ is equal to $.....$
  • A
    $1$
  • B
    $2$
  • C
    $3$
  • ✓
    $4$
✓

Answer

Correct option: D.
$4$
d
$|\vec{a}+\vec{b}+\vec{c}|^{2}=|\vec{a}|^{2}+|\bar{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \cdot \vec{b}+\overrightarrow{a \cdot c}+\vec{b} \cdot \vec{c})=3$

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

$\vec{a}(\vec{a}+\vec{b}+\vec{c})=|\vec{a}|+|\vec{a}+\vec{b}+\vec{c}| \cos \theta$

$\Rightarrow 1=\sqrt{3} \cos \theta$

$\Rightarrow \cos 2 \theta=-\frac{1}{3}$

$\Rightarrow 36 \cos ^{2} 2 \theta=4$

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