Question

Evaluate
\[∆ = \begin{vmatrix}0 & \sin \alpha & - \cos \alpha \\ - \sin \alpha & 0 & \sin \beta \\ \cos \alpha & - \sin \beta & 0\end{vmatrix}\]

Answer 1

Let 

\[∆ = \begin{vmatrix}0 & \sin\alpha & - \cos\alpha \\ - \sin\alpha & 0 & \sin\beta \\ \cos\alpha & - \sin\beta & 0\end{vmatrix}\]

\[∆ = \left( - 1 \right)^{1 + 1} 0 \left( 0 + \sin^2 \beta \right) + \left( - 1 \right)^{1 + 2} \sin\alpha\left( 0 - \sin\beta\cos\alpha \right) + \left( - 1 \right)^{1 + 3} \left( - \cos\alpha \right)\left( \sin\alpha\sin\beta - 0 \right) \left[ \text{ Expanding along } R_1 \right]\]
\[ = 0\left( 0 + \sin^2 \beta \right) - \sin\alpha\left( 0 - \sin\beta\cos\alpha \right) - \cos\alpha\left( \sin\alpha\sin\beta - 0 \right)\]
\[ = \sin\alpha\sin\beta\cos\alpha - \sin\alpha\sin\beta\cos\alpha\]
\[ = 0\]