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.