Question

The vector cos α cos β$\hat{i}$ + cos α sin β$\hat{j}$ + sin α$\hat{k}$ is a
(a) null vector
(b) unit vector
(c) constant vector
(d) none of these

Answer 1

(b) 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.