Question

If either vector $\vec {a}=\vec{0}\space or\space \vec{b}=\vec{0}$, then $\vec{a}.\vec{b}=0$.But the converse need not be true.Justify your answer with an example.

Answer 1

Consider $\vec{a}=2\hat{i}+4\hat{j}+3\hat{k}\space and\space \vec{b}=3\hat{i}+3\hat{j}-\hat{k}.$
Then,
$\vec{a}.\vec{b}=2.3+4.3+3(-6)=6+12-18=0$
We now observe that:
$|\vec{a}|=\sqrt{2^{2}+4^{2}+3^{2}}=\sqrt{29}$
$∴\vec{a}\ne\vec{0}$
$|\vec{b}|=\sqrt{3^{2}+3^{2}+(-6)^{2}}=\sqrt{54}$
$∴\vec{b}\ne\vec{0}$
Hence,the converse of the given statement need not be true.