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
Given: The vector \[\cos\alpha \cos\beta \hat{i} + \cos\alpha \sin \beta \hat{j} + \sin \alpha \hat{k} .\]
Then,
\[\left| \cos\alpha \cos \beta \hat{i} + \cos\alpha \sin\beta \hat{j} + \sin\alpha \hat{k} \right| = \sqrt{\cos^2 \alpha \cos^2 \beta + \cos^2 \alpha \sin^2 \beta + \sin^2 \alpha} = \sqrt{\cos^2 \alpha + \sin^2 \alpha} = 1\]
Hence, the given vector is a unit vector.