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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
Choose the correct answer
If $\theta$ is the angle between two vectors $\vec{\text{a}}\ \text{and}\ \vec{\text{b}},\ \text{then}\ \vec{\text{a}}\cdot\vec{\text{b}}\geq0$ only when,
  • A
    $0<\theta<\frac{\pi}{2}$
  • ✓
    $0\leq\theta\leq\frac{\pi}{2}$
  • C
    $0<\theta<\pi$
  • D
    $0\leq\theta\leq\pi$​​
✓

Answer

Correct option: B.
$0\leq\theta\leq\frac{\pi}{2}$
Let $\theta$ be the angle between two vectors $\vec{\text{a}}\ \text{and}\ \vec{\text{b}}.$Then, without loss of generality, $\vec{\text{a}}\ \text{and}\ \vec{\text{b}}$ are non-zero vectors so that $|\vec{\text{a}}|\ \text{and}\ \Big|\vec{\text{b}}\Big|$ are positive
It is known that $\vec{\text{a}}\cdot\vec{\text{b}}=|\vec{\text{a}}| \Big|\vec{\text{b}}\Big|\cos\theta.$
$\therefore\vec{\text{a}}\cdot\vec{\text{b}}\geq0$
$\Rightarrow|\vec{\text{a}}| \Big|\vec{\text{b}}\Big|\cos\theta\geq0$
$\Rightarrow\cos\theta\geq0\ \ \ \Big[\big|\vec{\text{a}}\big|\ \text{and}\ \Big|\vec{\text{b}}\Big|\ \text{are positive}\Big]$
$\Rightarrow0\leq\theta\leq\frac{\pi}{2}$
Hence, $\vec{\text{a}}.\vec{\text{b}}\geq0\ \text{when}\ 0\leq\theta\leq\frac{\pi}{2}.$
The correct answer is B.

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