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If $a_i^2 + b_i^2 + c_i^2 = 1,\,i = 1,2,3$ and $a_ia_j + b_ib_j +c_ic_j = 0$ $\left( {i \ne j,i,j = 1,2,3} \right)$ then the value of determinant $\left| {\begin{array}{*{20}{c}}
{{a_1}}&{{a_2}}&{{a_3}} \\
{{b_1}}&{{b_2}}&{{b_3}} \\
{{c_1}}&{{c_2}}&{{c_3}}
\end{array}} \right|$ is
$1/2$
$0$
$2$
$1$
Solution
$a_{i}^{2}+b_{i}^{2}+c_{i}^{2}=1 \quad i=\{1,2,3\}$
and $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=a_{1} a_{3}+b_{1} b_{3}+c_{1} c_{3}$
$=a_{2} a_{3}+b_{2} b_{3}+c_{2} c_{3}=0$
Now
Let $\Delta=\left|\begin{array}{lll}{a_{1}} & {a_{2}} & {a_{3}} \\ {b_{1}} & {b_{2}} & {b_{3}} \\ {c_{1}} & {c_{2}} & {c_{3}}\end{array}\right|$
so $\Delta^{2}=\left|\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right|$
so, $\Delta^{2}=1 \quad \Rightarrow \Delta \pm 1$
