Question

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

Answer 1

Consider $\overrightarrow{a}=2\hat{i}+4\hat{j}+3\hat{k}$ and $\overrightarrow{b}=3\hat{i}+3\hat{j}-6\hat{k}$.

Then,

$\overrightarrow{a}\cdot \overrightarrow{b}=2.3+4.3+3(-6)=6+12-18=0$

We now observe that:

$|\overrightarrow{a}|=\sqrt{2^2+4^2+3^2}=\sqrt{29}$

$\therefore \overrightarrow{a}\ne \overrightarrow{0}$

$|\overrightarrow{b}|=\sqrt{3^2+3^2+(-6{)}^2}=\sqrt{54}$

$\therefore \overrightarrow{b}\ne \overrightarrow{0}$

Hence, the converse of the given statement need not be true.