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Question
If \[\vec{a} . \vec{a} = 0 \text{ and } \vec{a} . \vec{b} = 0,\] what can you conclude about the vector \[\vec{b}\]
Solution
\[\text{ Given that } \vec{a} . \vec{a} = 0\]
\[ \Rightarrow \left| \vec{a} \right|^2 = 0\]
\[ \Rightarrow \left| \vec{a} \right| = 0 . . . \left( 1 \right)\]
\[\text{ Also, given that }\]
\[ \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta = 0........... (\text{ where } \theta \text{ is the angle between } \vec{a} \text{ and } \vec{b} )\]
\[ \Rightarrow 0 \left| \vec{b} \right| \cos \theta = 0 ............[\text{ From } (1)]\]
\[ \Rightarrow 0 = 0\]
\[\text{ So, it means that for any vector } \vec{b} , \text{ the given equation } \vec{a} . \vec{b} = 0 \text{ is satisfied }.\]
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