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Vector or Cross Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product3 Marks
Question
If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=7$ and $\vec{\text{a}}\times\vec{\text{b}}=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}},$ find the angle between $\vec{\text{a}}$ and $\vec{\text{b}}.$
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Answer

We know that, if $\theta$ be the angle between $\vec{\text{a}}$ and $\vec{\text{b}},$ then

$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\cos\theta\dots(1)$

And, $\vec{\text{a}}\times\vec{\text{b}}=|\vec{\text{a}}|\big|\vec{\text{b}}\big|.\sin\theta.\hat{\text{n}}$

$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{a}}|.\big|\vec{\text{b}}\big|.|\sin\theta|.|\hat{\text{n}}|$

$=|\vec{\text{a}}|\big|\vec{\text{b}}\big||\sin\theta|.1$ [Since, $\hat{\text{n}}$ is a unit vector]

$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{a}}|\big|\vec{\text{b}}\big|\sin\theta\dots(2)$

Given that, $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\vec{\text{a}}.\vec{\text{b}}$

$|\vec{\text{a}}|\big|\vec{\text{b}}\big|\sin\theta=|\vec{\text{a}}|.\big|\vec{\text{b}}\big|\cos\theta$

$\sin\theta=\cos\theta$

$\theta=\frac{\pi}{4}$

Angle between $\vec{\text{a}}$ and $\vec{\text{b}}=\frac{\pi}{4}$

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