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

$$\begin{gathered} \text{Let}\overrightarrow{a}\text{=}\left(cos\alpha cos\beta \right)\hat{i}+\left(cos\alpha sin\beta \right)\hat{j}+\left(sin\alpha \right)\hat{k} \\ \left|\overrightarrow{a}\right|\mathrm{=}\sqrt{co{s}^2\alpha co{s}^2\beta +co{s}^2\alpha si{n}^2\beta +si{n}^2\alpha } \\ \mathrm{=}\sqrt{co{s}^2\alpha \left(co{s}^2\beta +si{n}^2\beta \right)+si{n}^2\alpha } \\ \mathrm{=}\sqrt{co{s}^2\alpha \left(1\right)+si{n}^2\alpha } \\ \mathrm{=}\sqrt{co{s}^2\alpha +si{n}^2\alpha } \\ =\sqrt{1} \\ =1 \\ \text{So,}\overrightarrow{a}\text{is a unit vector.} \end{gathered}$$