QJEEAdv03Maace5: JEE Advanced 2003 Physics - Class 11 JEEAdv03Maace5 Solution: If are three non-coplanar unit vectors and are the angles…

If $\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,$ are three non-coplanar unit vectors and $\alpha ,\beta ,\gamma$ are the angles between $\overrightarrow u$ and $\overrightarrow v$ and $\overrightarrow w ,$ $\overrightarrow w$ and $\overrightarrow u$ respectively and $\overrightarrow x ,\overrightarrow y ,\overrightarrow z ,$ are unit vectors along the bisectors of the angles $\alpha ,\,\,\beta ,\,\,\gamma$ respectively. Prove that $\,\left[ {\overrightarrow x \times \overrightarrow y \,\,\overrightarrow y \times \overrightarrow z \,\,\overrightarrow z \times \overrightarrow x } \right] = {1 \over {16}}{\left[ {\overrightarrow u \,\,\overrightarrow v \,\,\overrightarrow w } \right]^2}\,{\sec ^2}{\alpha \over 2}{\sec ^2}{\beta \over 2}{\sec ^2}{\gamma \over 2}.$