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Question
What inference can you draw if \[\vec{a} \times \vec{b} = \vec{0} \text{ and } \vec{a} \cdot \vec{b} = 0 .\]
Solution
\[\text{ Given } :\]
\[\left| \vec{a} \times \vec{b} \right| = \vec{0} \]
\[ \Rightarrow \vec{a} = 0 \]
\[ \vec{b} =0\]
\[ \therefore \vec{a} \lVert \vec{b} \]
\[\text{ Also } , \]
\[ \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 0 \]
\[ \Rightarrow \vec{a} = \vec{0} or \vec{b} = \vec{0} or, \vec{a} \perp \vec{b} \]
\[\text{ But } \vec{a} \text { cannot be both perpendicular as well as parallel to } \vec{b} . \]
\[ \therefore \left| \vec{a} \right|=0\]
\[ \left| \vec{b} \right|=0\]
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