Question

If either $\overrightarrow{a}=\overrightarrow{0}\mathrm{or}\overrightarrow{b}=\overrightarrow{0},\mathrm{then}\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}.$ Is the converse true? Justify your answer with an example.

Answer 1

$$\begin{gathered} \mathrm{If}\overrightarrow{a}=\overrightarrow{0}\text{or}\overrightarrow{b}\text{=0, then}\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\sin \theta \hat{n}=\overrightarrow{0.} \\ \Rightarrow \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0} \\ \text{But the converse is not true as whenever}\overrightarrow{a}\text{×}\overrightarrow{b}\text{=}\overrightarrow{0}\text{, we cannot be sure that either}\overrightarrow{a}\text{=}\overrightarrow{0}\text{or}\overrightarrow{b}\text{=}\overrightarrow{0}\text{.} \\ \text{For example:} \\ \overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k} \\ \overrightarrow{b}=\hat{i}+2\hat{j}+3\hat{k} \\ \text{Here,} \\ \overrightarrow{a}\text{≠0} \\ \overrightarrow{b}\text{≠0} \\ \text{But}\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& 3\\ 1& 2& 3\end{array}\right| \\ =0\hat{i}+0\hat{j}+0\hat{k} \\ =\overrightarrow{0} \end{gathered}$$