Question

Let $\Delta =\left|\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \theta \sin \varphi & \sin \theta \cos \varphi & 0\end{array}\right|$, then

• A. $\Delta$ is independent of $\theta$
• B. $\Delta$ is independent of $\varphi$
• C. $\Delta$ is a constant
• D. $\Delta$ is dependent of $\varphi$

Answer 1

The correct option is B $\Delta$ is independent of $\varphi$

$△=\left[\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \theta \sin \varphi & \sin \theta \cos \varphi & 0\end{array}\right]$

expanding through $R_3$ we get

$-\sin \theta \sin \varphi (-{\sin }^2\theta \sin \varphi -{\cos }^2\theta \sin \varphi )-\sin \theta \cos \varphi (-{\sin }^2\theta \cos \varphi -{\cos }^2\theta \cos \varphi )$

$=\sin \theta {\sin }^2\varphi ({\sin }^2\theta +{\cos }^2\theta )+\sin \theta {\cos }^2\varphi ({\sin }^2\theta +{\cos }^2\theta )$

$=\sin \theta {\sin }^2\varphi +\sin \theta {\cos }^2\varphi =\sin \theta ({\sin }^2\varphi +{\cos }^{\varphi })$

$=\sin \theta$

Independent of $\varphi$