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 

Answer 1

unit vector 

\[\text{ Let } \vec{a} =\left( \text{ cos  }\alpha \text{ cos } \beta \right) \hat{i} +\left( \text{ cos }\alpha \text{ sin } \beta \right) \hat{j} +\left( \text{ sin }\alpha \right) \hat{k} \]
\[\left| \vec{a} \right|=\sqrt{cos^2 \alpha \text{ cos }^2 \beta + \text{ cos  }^2 \alpha \text{ sin}^2 \beta +  \text{ sin}^2 \alpha}\]
\[=\sqrt{co s^2 \alpha \left( co s^2 \beta + si n^2 \beta \right) + si n^2 \alpha}\]
\[=\sqrt{co s^2 \alpha\left( 1 \right) + si n^2 \alpha}\]
\[=\sqrt{co s^2 \alpha + si n^2 \alpha}\]
\[ = \sqrt{1}\]
\[ = 1\]
\[\text{ So }, \vec{a} \text{ is a unit vector }.\]