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Question
The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\] is a
Options
null vector
unit vector
constant vector
None of these
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Solution
unit vector
\[\text{ Let } \vec{a} =\left( \text{ cos }\alpha \text{ cos } \beta \right) \hat{i} +\left( \text{ cos }\alpha \text{ sin } \beta \right) \hat{j} +\left( \text{ sin }\alpha \right) \hat{k} \]
\[\left| \vec{a} \right|=\sqrt{cos^2 \alpha \text{ cos }^2 \beta + \text{ cos }^2 \alpha \text{ sin}^2 \beta + \text{ sin}^2 \alpha}\]
\[=\sqrt{co s^2 \alpha \left( co s^2 \beta + si n^2 \beta \right) + si n^2 \alpha}\]
\[=\sqrt{co s^2 \alpha\left( 1 \right) + si n^2 \alpha}\]
\[=\sqrt{co s^2 \alpha + si n^2 \alpha}\]
\[ = \sqrt{1}\]
\[ = 1\]
\[\text{ So }, \vec{a} \text{ is a unit vector }.\]
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