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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA5 Marks
Question
If either $\vec{a}=\vec{0}\ \ \text{or}\ \ \vec{b}=\vec{0},\ \ \text{then}\ \ \vec{a}\times\vec{b}=\vec{0}.$ Is the converse true? Justify your answer with an example.
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Answer

$\text{Given:}\ \ \text{Either}\ \vec{a}=\vec{0}\ \ \ \text{or}\ \ \ \vec{b}=\vec{0}$ $\therefore\ \ \Big|\vec{a}\Big|=\Big|\vec{0}\Big|=0\ \ \text{or}\ \ \ \Big|\vec{b}\Big|=\Big|\vec{0}\Big|=0\ \ \ ........\text{(i)}$ $\therefore\ \Big|\vec{a}\times\vec{b}\Big|=\Big|\vec{a}\Big|\Big|\vec{b}\Big|\sin\theta=0\ $ $\Rightarrow\ \ 0.\sin\theta=0\ \ \big[\text{Using eq.(i)}\big]$ $\therefore\ \vec{a}\times\vec{b}=\vec{0}\ \ \ \ \text{[By definition of zero vector]}$ But the converse is not true. $\text{Let}\ \ \ \vec{a}=\hat{i}+\hat{j}+\hat{k}\ $ $\Rightarrow\ \ \big|\vec{a}\big|=\sqrt{1+1+1}=\sqrt{3}\neq0$ $\therefore\ \ \ \ \ \ \vec{a}\ \text{is a non-zero vector.}$ $\text{Let}\ \ \ \vec{b}=2\Big(\hat{i}+\hat{j}+\hat{k}\Big)=2\hat{i}+2\hat{j}+2\hat{k}\ $ $\Rightarrow\ \ \Big|\vec{b}\Big|=\sqrt{4+4+4}=\sqrt{12}=2\sqrt{3}\neq0$ $\therefore\ \ \ \ \ \vec{b}\ \text{is a non-zero vector.}$ $\text{But}\ \ \ \ \ \ \vec{a}\times\ \vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\1&1&1\\2&2&2\end{vmatrix}$ $\text{Taking 2 common from R}_{3}=\vec{a}\times\ \vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\1&1&1\\1&1&1\end{vmatrix}=\vec{0}\ \ \ $ $\big[\because\text{R}_2\ \text{and R}_{3}\ \text{are identical}\big]$

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