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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA2 Marks
Question
$\text{Given that}\ \vec{a}\cdot\vec{b}=0\ \text{and}\ \vec{a}\times\vec{b}=\vec{0}.$ What can you conclude about the vectors $\vec{a}\ \text{and}\ \vec{b}$?
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Answer

$\text{Given:}\ \ \vec{a}.\vec{b}=0\ \Rightarrow\ \ \big|\vec{a}\big|.\big|\vec{b}\big|\cos\theta=0$

$\therefore\ \ \big|\vec{a}\big|=0\ \ \text{or} \ \big|\vec{b}\big|=0$$\ \text{or}\ \cos\theta=0\ \ \Rightarrow\ \ \theta=90^\circ$

$\Rightarrow \ \ \vec{a}=0\ \ \text{or}\ \ \vec{b}=0\ \ \text{or}$ $\ \ \text{vector}\ \vec{a}\ \text{is perpendicular to}\ \vec{b}.\ \ \ \ ......\text{(i)}$

$\text{Again, given}\ \vec{a}\times\vec{b}=0\ \Rightarrow\ \big|\vec{a}\times\vec{b}\big|=0$ $\ \Rightarrow\ \ \big|\vec{a}\big|.\big|\vec{b}\big|\sin\theta=0$

$\therefore\ \ \big|\vec{a}\big|=0\ \ \text{or}\ \ \big|\vec{b}\big|=0\ \ \text{or}\ \ $ $\sin\theta=0\ \ \Rightarrow\ \theta=0^\circ$

$\Rightarrow\ \ \vec{a}=0\ \ \text{or}\ \ \vec{b}=0 \ \ \text{or}\ \ $ $\text{vector}\ \vec{a}\ \text{and}\ \vec{b}\ \text{are collinear or parallel.}\ \ \ \ \ ...\text{(ii)}$

Since, vectors $\vec{a}\ \&\ \vec{b}$ are perpendicular to each other as well as parallel are not possible. ...(iii)

Therefore, form eq. (i), (ii) and (iii), $\ \text{either}\ \vec{a}=\vec{0}\ \ \ \text{or}\ \vec{b}=\vec{0}$

$\therefore\ \ \vec{a}.\vec{b}=0 \ \ \text{and}\ \ \vec{a}\times\vec{b}=0$

 

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