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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra2 Marks
Question
If either $\vec{a}=\vec {0}$ or $\vec{b}=\vec{0}$, then $\vec{a} \times \vec{b}=\vec 0$. Is the converse true? Justify your answer with an example.
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Answer

Firstly let us consider, Take any parallel non-zero vectors so that
$\vec{a} \times \vec{b}=\vec{0}$
Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}$, $\vec{b}=4 \hat{i}+6 \hat{j}+8 \hat{k}$
Then,
$\vec{a} \times \vec{b}$ = $\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 4 & 6 & 8\end{array}\right|$ = $\hat{i}(24-24)-\hat{j}(16-16)+\hat{k}(12-12)$ = $0 \hat{i}+0 \hat{j}+0 \hat{k}=\vec{0}$
Now, it's seen that $|\vec{a}|=\sqrt{2^{2}+3^{2}+4^{2}}=\sqrt{29}$
$\therefore \vec{a} \neq \vec{0}$
$|\vec{b}|=\sqrt{4^{2}+6^{2}+8^{2}}=\sqrt{116}$
$\therefore \vec{b} \neq \vec{0} $
Thus, the converse of the given statement need not be true.

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