Question

$$\begin{gathered} Q.If\underset{}{\sum {\cos }^2{\alpha }_1=\sum {\cos }^2{\beta }_1=\sum {\cos }^2{\gamma }_1=1and} \\ \sum \cos {\alpha }_1\cos {\beta }_1=\sum \cos {\beta }_1\cos {\gamma }_1=\sum \cos {\gamma }_1\cos {\alpha }_1=0 \\ Then,findthevalueof \\ \left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right| \end{gathered}$$

Answer 1

$$\begin{gathered} \left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right| \\ Consider \\ \left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right|\left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right| \\ Multiplyrowbyrow \\ =\left|\begin{array}{ccc}{\cos }^2{\alpha }_1+{\cos }^2{\alpha }_2+{\cos }^2{\alpha }_3& \cos {\alpha }_{1}cos{\beta }_1+\cos {\alpha }_{2}cos{\beta }_2+\cos {\alpha }_{3}cos{\beta }_3& \cos {\alpha }_{1}cos{\gamma }_1+\cos {\alpha }_{2}cos{\gamma }_2+\cos {\alpha }_{3}cos{\gamma }_3\\ \cos {\alpha }_{1}cos{\beta }_1+\cos {\alpha }_{2}cos{\beta }_2+\cos {\alpha }_{3}cos{\beta }_3& {\cos }^2{\beta }_1+{\cos }^2{\beta }_2+{\cos }^2{\beta }_3& \cos {\beta }_{1}cos{\gamma }_1+\cos {\beta }_{2}cos{\gamma }_2+\cos {\beta }_{3}cos{\gamma }_3\\ \cos {\alpha }_{1}cos{\gamma }_1+\cos {\alpha }_{2}cos{\gamma }_2+\cos {\alpha }_{3}cos{\gamma }_3& \cos {\beta }_{1}cos{\gamma }_1+\cos {\beta }_{2}cos{\gamma }_2+\cos {\beta }_{3}cos{\gamma }_3& {\cos }^2{\gamma }_1+{\cos }^2{\gamma }_2+{\cos }^2{\gamma }_3\end{array}\right| \\ =\left|\begin{array}{ccc}\sum {\cos }^2{\alpha }_1& \sum \cos {\alpha }_{1}cos{\beta }_1& \sum \cos {\alpha }_{1}cos{\gamma }_1\\ \sum \cos {\alpha }_{1}cos{\beta }_1& \sum {\cos }^2{\beta }_1& \sum \cos {\beta }_{1}cos{\gamma }_1\\ \sum \cos {\alpha }_{1}cos{\gamma }_1& \sum \cos {\beta }_{1}cos{\gamma }_1& \sum {\cos }^2{\gamma }_1\end{array}\right| \\ =\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right| \\ =1 \\ \Rightarrow {\left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right|}^2=1 \\ \left|\begin{array}{ccc}\cos {\alpha }_1& \cos {\alpha }_2& \cos {\alpha }_3\\ \cos {\beta }_1& \cos {\beta }_2& \cos {\beta }_3\\ \cos {\gamma }_1& \cos {\gamma }_2& \cos {\gamma }_3\end{array}\right|=\pm 1 \\ Note:\hspace{0.17em}Wecannotpreciselytellthatwhetheritis-1or1becausemuchoftheinformationisnotprovided \end{gathered}$$