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Question
If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible?
Options
\[\sqrt{3}\]
\[\sqrt{3}/2\]
\[1/\sqrt{2}\]
−1/2
Solution
\[\sqrt{3}\]
\[\text{ It is given that } \vec{a} \text{ and } \vec{b} \text{ are unit }vectors.\]
\[ \Rightarrow \left| \vec{a} \right| = \left| \vec{b} \right| = 1\]
\[\text{ Now },\]
\[ \vec{a} . \vec{b} \]
\[ = \left| \vec{a} \right| \left| \vec{b} \right| \cos \theta\]
\[ = \left( 1 \right) \left( 1 \right) \cos \theta\]
\[ = \cos \theta\]
\[\text{ The range of }\cos \theta \text{ is }\left[ - 1, 1 \right].\]
\[ \therefore \sqrt{3}\text{ is not a possible value of }\cos\theta \text{ as it is greater than } 1.\]
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