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Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product3 Marks
Question
If either $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0},$ then $\vec{\text{a}}\times\vec{\text{b}}=\vec{0}.$ is the converse true? justify your answer with an example.
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Answer

If $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0},$ then $|\vec{\text{a}}|\big|\vec{\text{b}}\big|\sin\theta\text{ n}=\vec{0}.$
$\Rightarrow\vec{\text{a}}\times\vec{\text{b}}=\vec{0}$
But the converse is not true as whenever $\vec{\text{a}}\times\vec{\text{b}}=\vec{0},$ we cannot be sure that either $\vec{\text{a}}=\vec{0}$ or $\vec{\text{b}}=\vec{0}.$
For exampale:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}+3\vec{\text{k}}$
$\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\vec{\text{k}}$
Here,
$\vec{\text{a}}\neq0$
$\vec{\text{b}}\neq0$
But  $\vec{\text{a}}\times\vec{\text{b}}=\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&2&3\\1&2&3 \end{vmatrix}$
$=0\hat{\text{i}}+0\hat{\text{j}}+0\hat{\text{k}}$
$=\vec{0}$

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