Let $\overrightarrow{{\mathrm{a}}_{\mathrm{r}}}={\mathrm{x}}_{\mathrm{r}}\widehat{\mathrm{i}}+{\mathrm{y}}_{\mathrm{r}}\widehat{\mathrm{j}}+{\mathrm{z}}_{\mathrm{r}}\widehat{\mathrm{k}}$, r = 1, 2, 3 be three mutually perpendicular unit vectors, then the value of $\left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{x}}_2& {\mathrm{x}}_3\\ {\mathrm{y}}_1& {\mathrm{y}}_2& {\mathrm{y}}_3\\ {\mathrm{z}}_1& {\mathrm{z}}_2& \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{z}}_3\end{array}\right|$ is equal to

$\mathrm{\Delta }=\left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{x}}_2& {\mathrm{x}}_3\\ {\mathrm{y}}_1& {\mathrm{y}}_2& {\mathrm{y}}_3\\ {\mathrm{z}}_1& {\mathrm{z}}_2& {\mathrm{z}}_3\end{array}\right|=\left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|$

$\begin{array}{rcl}\therefore {\mathrm{\Delta }}^2& =& \left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|\left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|\\ & =& \left|\begin{array}{ccc}{\mathrm{x}}_1^2+{\mathrm{y}}_1^2+{\mathrm{z}}_1^2& {\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{y}}_1{\mathrm{y}}_2+{\mathrm{z}}_1{\mathrm{z}}_2& {\mathrm{x}}_1{\mathrm{x}}_3+{\mathrm{y}}_1{\mathrm{y}}_3+{\mathrm{z}}_1{\mathrm{z}}_3\\ {\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{y}}_2{\mathrm{y}}_1+{\mathrm{z}}_2{\mathrm{z}}_1& {\mathrm{x}}_2^2+{\mathrm{y}}_2^2+{\mathrm{z}}_2^2& {\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{y}}_2{\mathrm{y}}_3+{\mathrm{z}}_2{\mathrm{z}}_3\\ {\mathrm{x}}_3{\mathrm{x}}_1+{\mathrm{y}}_3{\mathrm{y}}_1+{\mathrm{z}}_3{\mathrm{z}}_1& {\mathrm{x}}_2{\mathrm{x}}_3+{\mathrm{y}}_2{\mathrm{y}}_3+{\mathrm{z}}_2{\mathrm{z}}_3& {\mathrm{x}}_3^2+{\mathrm{y}}_3^2+{\mathrm{z}}_3^2\end{array}\right|\\ & =& \left|\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\end{array}\right|=1\Rightarrow \mathrm{\Delta }=\pm 1\end{array}$