Question

If \[\vec{a,} \vec{b}\] are two vectors, then write the truth value of the following statement: 
\[\left| \vec{a} \right| = \left| \vec{b} \right| \Rightarrow \vec{a} = \vec{b}\]

Answer 1

False.
We cannot say 
\[\left| \vec{a} \right| = \left| \vec{b} \right| \Rightarrow \vec{a} = \vec{b}\] 
Consider an example,
\[\vec{a} = i + \sqrt{3}j\text{ and }\vec{b} = \sqrt{2}i + \sqrt{2}j\]
\[\left| \vec{a} \right| = \sqrt{1^2 + \left( \sqrt{3} \right)^2} = 2 \text{ and }\left| \vec{b} \right| = \sqrt{\left( \sqrt{2} \right)^2 + \left( \sqrt{2} \right)^2} = 2\]
\[\text{ Thus, }\left| \vec{a} \right| = \left| \vec{b} \right|\text{ but }\vec{a} \neq \vec{b} \]