Explanation

(c)
We have, $a_{i}^{2} + b_{i}^{2} + c_{i}^{2} = 1$
and $a_{i}a_{j} + {b_{i}b}_{j} + {c_{i}c}_{j} = 0$ for ($i = 1,2,3)$
$\therefore\ \left| \begin{matrix}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{matrix} \right|^{2} = \left| \begin{matrix}
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2} \\
a_{3} & b_{3} & c_{3}
\end{matrix} \right|\left| \begin{matrix}
a_{1} & b_{1} & c_{1} \\
a_{2} & b_{2} & c_{2} \\
a_{3} & b_{3} & c_{3}
\end{matrix} \right|$
$= \left| \begin{matrix}
a_{1}^{2} + b_{1}^{2} + c_{1}^{2} & a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} & a_{1}a_{3} + b_{1}b_{3} + c_{1}c_{3} \\
a_{2}a_{1} + b_{2}b_{1} + c_{2}c_{1} & a_{2}^{2} + b_{2}^{2} + c_{2}^{2} & a_{1}a_{3} + b_{1}b_{3} + c_{1}c_{3} \\
a_{3}a_{1} + b_{3}b_{1} + c_{3}c_{1} & a_{3}a_{2} + b_{3}b_{2} + c_{3}c_{2} & a_{3}^{2} + b_{3}^{2} + c_{3}^{2}
\end{matrix} \right|$
$= \left| \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{matrix} \right| = 1$