Question

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

Answer 1

If a = 0 or b = 0, then surely $a\times b=0$
However, the converse is not true i.e., if $a\times b=0$, then a = 0 or b = 0 may not hold.
We know $a\times b=absin\theta \hat{n}$, if a and b are parallel, then $\theta =0^{\circ }$ and $sin0=0$.
$a\times b=0$
As an example, consider the vectors $a=\hat{i},b=\hat{i}$, then $a\ne 0$ and also $b\ne 0$.
But $a\times b=\hat{i}\times \hat{i}=0$
Alternate method
Let $a=\hat{i}+\hat{j}+\hat{k}$ and $b=4\hat{i}+4\hat{j}+4\hat{k}$
$\therefore |a|=\sqrt{1^2+1^2+1^2}=\sqrt{1+1+1}=\sqrt{3}\ne 0$
$|b|=\sqrt{4^2+4^2+4^2}=\sqrt{16+16+16}=\sqrt{48}=4\sqrt{3}\ne 0$
Now, $a\times b=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 1\\ 4& 4& 4\end{array}\right|=(4-4)\hat{i}-(4-4)\hat{j}+(4-4)\hat{k}=0$
Thus, $a\times b=0$, when $|a|\ne 0$ and $|b|\ne 0$.

Note: If $a\times b=0$, then it is not necessary that either |a| = 0 or |b| = 0