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Vector or Cross Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product2 Marks
Question
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are two vectors such that $\big|\vec{\text{a}}.\vec{\text{b}}\big|=\big|\vec{\text{a}}\times\vec{\text{b}}\big|,$ write the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$
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Answer

Let $\theta$ be the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$
We know
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\big|\sin\theta\big|$
$\big|\vec{\text{a}}.\vec{\text{b}}\big|=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\big|\cos\theta\big|$
Now,
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\big|\vec{\text{a}}.\vec{\text{b}}\big|$ (Given)
$\Rightarrow|\vec{\text{a}}|\big|\vec{\text{b}}\big||\sin\theta|=|\vec{\text{a}}|\big|\vec{\text{b}}\big||\cos\theta|$
$\Rightarrow|\sin\theta|=|\cos\theta|$
$\Rightarrow\theta=\frac{\pi}{4}$

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