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Vector or Cross Product — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector or Cross Product4 Marks
Question
If $|\vec{\text{a}}|=13,\big|\vec{\text{b}}\big|=5$ and $\vec{\text{a}}.\vec{\text{b}}=60,$ then find $\big|\vec{\text{a}}\times\vec{\text{b}}\big|.$
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Answer

We know that, if $\theta$ is angle between $\vec{\text{a}}$ and $\vec{\text{b}},$
$\vec{\text{a}}.\vec{\text{b}}=|\vec{\text{a}}||\vec{\text{b}}|\cos\theta$
$60=13.5.\cos\theta$
$\cos\theta=\frac{60}{65}$
$\cos\theta=\frac{12}{13}$
$\sin^2\theta=1-\cos^2\theta$
$=1-\Big(\frac{12}{13}\Big)^2$
$=1-\frac{144}{169}$
$=\frac{169-144}{169}$
$=\frac{25}{169}$
$\sin\theta=\pm\sqrt{\frac{25}{169}}$
$=\pm\frac{5}{13}$
$|\sin\theta|=\frac{5}{13}$
We know that,
$\vec{\text{a}}\times\vec{\text{b}}=|\vec{\text{a}}|.\big|\vec{\text{b}}\big|.\sin\theta.\hat{\text{n}}$
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{a}}|.\big|\vec{\text{b}}\big|.|\sin\theta|.|\hat{\text{n}}|$
$=13.5.\frac{5}{13}.1$ [Since, $\hat{\text{n}}$ is aunit vector]
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=25$

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